First things first: quantum mechanics is the bomb. I love this stuff.
Like Frère said, these topics are pretty abstract, so I'm here mostly to give you a second take on it and some more metaphors in case you don't like Frère's description. I find that more than one explanation is often helpful to my understanding, so here goes nothing.
The Double-Slit Experiment
Frère already described the experiment itself pretty well, so I'm just going to clarify and expand on it. We'll go back to his "shooting paintballs at a fence" analogy because it's a good one for this experiment.
Imagine the fence has only one small, vertical slit. What would you expect to see? Well, there should be one vertical stripe of paint on the back wall. If you really are using paintballs, that's what you'd see. Instead, shine a collimated flashlight directly at the fence. What do you see on the back wall? It's not just a slit, is it? It spreads out horizontally, getting dimmer as you go left or right. This is called a diffraction pattern, and it's what we intuitively expect light to do--it spreads out. If light were made up of particles like paintballs--even very, very small ones--you'd see a single vertical stripe on the back wall. The diffraction pattern, however, is something you expect from a wave, which tells us that light acts more like a wave in this situation.
The double-slit version is very similar, but more convincing. Shooting paintballs should give you two vertical slits, which it does. You would expect that shining a flashlight would give you two versions of the single-slit pattern with the horizontal spreading. There's a space on the back wall, however, in between the two slits, where the two patterns overlap, so you get an interference pattern, just like you would with any other wave. It's hard to see with walls and flashlights, but scientists have done it on very small scales. Again, if light were made up of very small photon paintballs, you wouldn't see that pattern, but we do. The picture below (taken from Wikipedia) might help.
Great! Now let's complicate things.
In 1905, Einstein explained the photoelectric effect with his concept of photons, effectively particles of light, and receives a Nobel Prize for it. Cue confusion. I thought we had this nailed down with the double-slit thing. We showed that light is a wave, and we can even measure its wavelength, but now you're telling me it acts like a particle! So which is it?
A Double-Slit Experiment Variation
Scientists then tried to devise an experiment to push the debate one way or another. In short, they asked, "What would happen if you shot just one photon at the slits?"
This is one really important variant of the double-slit experiment. Instead of shooting a bunch of paintballs at the fence, you only shoot one. The paintball should go through one of the slits and make a single paint splotch on the back wall, right? It turns out that if you shoot one photon at our double-slit contraption, that's what happens. You get one dot on the "back wall," or the detector. Great! When we boil it down to individual photons, they act like individual particles. Problem solved!
Well, not quite. Now shoot another paintball. It goes through one of the two slits in the fence and you get one more splotch. Same thing happens if you shoot one more photon. So, if we keep shooting individual paintballs, we'll see two vertical stripes of paint emerge on the back wall. The same thing should happen with the photons, right?
When you shoot individual photons, one by one, you eventually get the interference pattern back again. What gives? It's as if each of those individual particles adds up to a wave, but we designed the experiment so that each of the particles is exactly the same, so that can't be right. What if--and I'm going to say something crazy now--what if each of the individual particles is a wave?
Now that raises all sorts of interesting questions. How can something be both a particle and a wave? Are there actually two different "parts" to light? Why don't paintballs act like that? What on earth is going on?
The other implication, as Frère pointed out, is that our mode of measurement changes the result of the experiment. Shoot one photon paintball, and you get one spot on the back wall. That indicates particle behavior, but that's also the nature of the wall. In order to see where the paintball went, we have to stop it from moving; similarly, in order to detect a photon, we have to absorb it, which causes it to, well, stop being a photon. This is what we mean when we say that the act of observing the experiment changes the outcome.
Another metaphor: imagine that you want to study an insect--a housefly, perhaps. In this metaphor, the only way to see and study the fly is to catch it because it moves too fast on its own for you to get any good observations. You decide to catch it by putting up flypaper. That way, the fly sticks to the paper when it runs into it, but you haven't touched it, which means, in theory, that you haven't affected it. Here' the catch: however good your observations of the immobile fly is, they can't tell you anything about the pattern in which it flew to get there.
That fly represents our photon. Once it's absorbed, or "stuck" to the detector, it can't behave like a photon any longer. All we can observe is the "stuck" photon, which isn't really a photon anymore. Perhaps an actual photon moves in a wave pattern, but we can't observe that because we can't see it moving. In other words, we don't know anything about the pattern in which it flew to get to the detector.
Now back to the fly. If you were to leave your flypaper up for a long time, eventually you'd catch a lot of flies. Since they're flies, they'd probably be arranged somewhat randomly on the paper. Using that information, you can reasonably conclude that they fly somewhat randomly. When we collect a large number of photons on our detector, we're taking measurements in aggregate. It just so happens that the pattern is not random, and that it's consistent with wave behavior. One photon can't tell you anything about how it got there, but lots of photons can.
When you hear the phrase, "observing the experiment changes its outcome" on its own, it sounds downright mystical. My description is definitely not perfect, but this is how I understand the phenomena, and I hope this explanation clears up what scientists mean when they use this phrase.
The next couple sections will lead me to the article you asked about, but they also set up an important discussion on scaling, which sums up my beef with the article's conclusions. Bear with me, please.
Another Historical Interlude and Another Variation
There's another variation on the double-slit experiment that's still being explored. What happens if you use something other than photons? It's a great idea. After all, we know know that, say, an electron is a bona fide particle. It has mass and everything! What happens if you do the double-slit experiment with electrons?
Claus Jönsson did just that in 1961, and curiously enough, he got the same results with electrons as Young did with light. Cue Italian physicists Merli, Missiroli, and Pozzi, who performed the experiment using single electrons in 1974, with the same results. Wait, what? So now electrons are waves, too?
Additionally, the questions that scientists are asking have evolved. Wheeler's delayed-choice thought experiment, another variation on the double-slit experiment, does not explicitly ask, "Is this photon a particle or a wave?" Instead, it asks, "When did the photon 'decide' whether it was going to behave like a particle or a wave?" and "Can the photon 'detect' the experimental apparatus and adjust its behavior accordingly?" At the time Wheeler proposed his experiment, many scientists believed that a photon is either a wave or a particle, but that it cannot be both at the same time. Wheeler himself seemed clear on the idea of wave-particle duality; he felt that quantum phenomena are "intrinsically undefined until the moment they are measured." It wasn't until long after its inception, however, that scientists were able to actually create the experiment and test Wheeler's hypotheses.
Present Day: Dr. Truscott's Experiment
The article you linked to talks about an actual scientific paper, "Wheeler's delayed-choice gedanken experiment with a single atom," which you can read here, if you are so inclined. Its contribution to the field of physics is related to that second variation of the double-slit experiment. So far in the history of physics, the experiment has been effectively performed using a beam of photons, a beam of electrons, a single electron, and a single photon, roughly in that order. This paper, however, scales up by using a single, metastable helium atom.
The paper sums up its conclusions with this: "Our experiment confirms Bohr's view that it does not make sense to ascribe the wave or particle behavior to a massive particle before the measurement takes place." In more detail, it concludes in its last paragraph that
"Wheeler’s thought experiment is important since it tries to force a classical view of reality on to a quantum system. If one holds the view that to observe interference at the detector the photon must have traversed both arms (as a wave) of the interferometer (and conversely that the lack of interference unambiguously demonstrates the photon has traversed a single arm (as a particle)) then the ‘delayed’ choice creates a conundrum. In this picture, the choice of detection (delayed until after the photon has passed the first beamsplitter) is correlated with observing interference or no interference—and thus it seems that a future event (the method of detection) causes the photon to decide its past. If such a perspective seems untenable with a fast-moving massless photon, then our experiment, which uses a slow-moving massive helium atom (and thus is closer to our classical notions), makes this view of reality seem even more unlikely."
Roughly translated, this says that Wheeler was probably right. When they placed the detector behind the slits (or the fence, in our paintball analogy), they saw an interference pattern, but when they placed the detector in front of them, no interference pattern showed up. Because an atom cannot "know" if, when, or where it will be detected, it makes no sense to say that it "decided" at any point how it was going to behave. In essence, because the future cannot affect its past, the atom exists in both "wave" and "particle" states at one time.
The Daily Mail Article
If my understanding of the scientific paper is correct, then the reporter of the Daily Mail article you linked to, Ellie Zolfagharifard, got things a bit backward. She reports,
If you choose to believe that the atom really did take a particular path or paths then you have to accept that a future measurement is affecting the atom's past, said Truscott.
'The atoms did not travel from A to B. It was only when they were measured at the end of the journey that their wave-like or particle-like behaviour was brought into existence,' he said.
Those things are true, according to the paper. What Zolfagharifard seems to have missed is the last portion of the quote above from the paper, namely the "makes this view of reality seem even more unlikely [emphasis mine]." The paper assumes that the atom has to have a fixed state--particle or wave--but then goes on to conclude that that assumption can't be correct, which the Daily Mail article leaves out entirely.
Scaling and Media Exaggeration
Dr. Truscott is quoted in the article, saying, "It proves that measurement is everything. At the quantum level, reality does not exist if you are not looking at it." Yes, the statement, "reality does not exist if you are not looking at it" sounds suspect, but you may notice that the quote does not claim, as the title of the article does, that "Your entire life is an ILLUSION," or "the world doesn’t exist until we look at it."
There's a big difference between reality at the quantum level and "the world." Just because something is true on an atomic scale does not mean that it is true on a macroscopic scale. I mean, sometimes it does, but more often than not, it doesn't. Physics is weird like that. Dr. Truscott does, in fact, qualify his statement with, "At the quantum level," but that's conveniently glossed over without any more discussion.
Here's a metaphor that conveys a little bit of why scaling is important. Imagine a car on a highway. Every little dip and bump and inconsistency in the road technically affects its travel route and time, but the deviations are so small compared to your intended route--say, across town--that they don't matter. They're negligible. If you and your car shrunk so that the car was only a millimeter long, those bumps would matter a whole lot more. The route you take in and around them will affect your travel time significantly.
Atoms are a little like the bumps in the road. They experience all sorts of vibrations and rotations and energetic movements, but those movements are so small compared to what we experience in our day-to-day lives (or the entire road trip, if you will) that they don't matter very much. And yet, despite not mattering very much individually, they add up to, well, everything. The experimenters conclude one thing about your travel time at the bumps-in-the-road level, but the article tries to use that to report your across-town travel time. They're just not the same thing. Dr. Truscott made a conclusion about one atom in his experiment, but to scale that up and apply it to all of reality is kind of ridiculous.
The media are simply exacerbating the problem, but I think there's more than one reason why. Yes, the media often skews our perception of things, sometimes intentionally, to gain and retain readers. That's just something they do. The other problem is that journalists are not specialists. It's not just you that can't easily decipher the crazy-dense scientific articles. Even with interviews and explanations, reporters don't always understand what they're reporting, especially in science and technology. It's not that they don't make an effort, because I'm sure many of them do, but these are not easy topics to understand. Period. I can all but guarantee that it wasn't easy for even the people doing the study to understand. I think it's that, combined with the media's penchant for exaggeration and extrapolation, that makes a reader's job so difficult.
My thoughts on the article: it isn't really accurate, but hey, they tried.
To summarize the science: A "particle", be it atom or molecule or photon, behaves like it does. We're trying to ascribe to it behaviors that we understand, but our description is fundamentally flawed because the particle doesn't behave in a way we understand in the first place. A photon acts like a photon and exhibits "photon behavior," but we don't know how to measure that, so we have to use our flawed models (like particles and waves) to figure out what that means. This is especially important in quantum mechanics because the scientific models we're used to--things like Newton's laws--are macroscopic. They don't necessarily apply at such small scales, which is why scaling is important.
In conclusion: the media isn't very good at accurate scientific reporting. Probably because science is hard and complicated.