To add to that, it is a relatively common classroom experiment to show trails in gas left by muons from cosmic radiation. These muons are travelling at about 99.94% of the speed of light, which is quite fast but the distance from the upper atmosphere where they originate to the classroom is long enough that it takes the muon several of its half-lives to reach the clasroom - by our measurement of time, at least. We should expect them to have decayed before the reach the classroom, but they don't!
By doing the same experiment at multiple elevations we can see that the rate of muon decay is much lower than non-relativistic theories would suggest. However, if time dilation due to their large speed is taken into account then we get that the muons 'experience' a much shorter trip from their point of view - sufficiently short that they don't decay! That they have reached the classroom is evidence (given a bunch of other knowledge about decay and formation of muons) that is easily observed for time dilation.
Also! Time dilation is surprisingly easy to derive. I recommend that you attempt to derive it yourself if you haven't already! I give you this starting point:
A) The speed of light is constant and independent of observers
B) A simple way to analyze time is to consider a very simple clock: two mirrors facing towards each other with a photon bouncing back and forth between the two. The cycles of the photon denotes the passage of time.
C) What if the clock is moving?
D) Draw a diagram
In response to falenas108's "Ask an X" thread. I have a PhD in experimental particle physics; I'm currently working as a postdoc at the University of Cincinnati. Ask me anything, as the saying goes.
This is an experiment. There's nothing I like better than talking about what I do; but I usually find that even quite well-informed people don't know enough to ask questions sufficiently specific that I can answer any better than the next guy. What goes through most people's heads when they hear "particle physics" is, judging by experience, string theory. Well, I dunno nuffin' about string theory - at least not any more than the average layman who has read Brian Greene's book. (Admittedly, neither do string theorists.) I'm equally ignorant about quantum gravity, dark energy, quantum computing, and the Higgs boson - in other words, the big theory stuff that shows up in popular-science articles. For that sort of thing you want a theorist, and not just any theorist at that, but one who works specifically on that problem. On the other hand I'm reasonably well informed about production, decay, and mixing of the charm quark and charmed mesons, but who has heard of that? (Well, now you have.) I know a little about CP violation, a bit about detectors, something about reconstructing and simulating events, a fair amount about how we extract signal from background, and quite a lot about fitting distributions in multiple dimensions.