From a high school student: I read somewhere that time gets all messed up because of the theory of relativity or something. What's up with that?

Well, the theory of relativity was a new physics idea developed in the early 1900's by Albert Einstein, radically generalizing ideas from electromagnetism at the time (e.g. invariance of Maxwell's equations in different reference frames). It says that the old laws of kinematics and dynamics - motion and forces - by Newton and Kepler and all those guys who we learn about in school are in fact only approximations to what really happens. The equations we learned early on, like "distance equals speed multiplied by time", work just fine for daily life things like cars and airplanes and bowling balls, and are still important for those describing things. But when we deal with objects that move at close to the speed of light, or that are near large gravity fields, our traditional equations and experiences are way off, time and space don't act like you would expect, and we need to use more complicated equations to describe the motion.

In this diagram we can see how the curved blue line is approximated pretty well by the straight red line for about half the graph. In the same way, the traditional physics of Newton only approximate what's predicted by the theory of relativity at slower speeds.

An example of this strange relativity behavior is on the GPS satellites which orbit the Earth and allow our handheld GPS receivers to find our location. You may have seen these gadgets in cars and boats. The GPS satellites carry very fast-ticking, very accurate clocks on board, and radio their times down to us on the ground as part of the location-finding process. Now it may sound weird since we don't see it in daily life, but relativity effects actually cause time to flow differently when you move at a high speed or are in the presence of a large gravitational field. Those effects in fact happen to us all the time on the Earth here but they're so small that we don't notice them since we walk and drive around pretty slowly, and Earth's gravitational field is strong but not ultra-strong. And yet, it turns out these GPS satellites move fast enough in their orbits, and their clocks are so accurate and work on such a fine scale, that we actually do notice these small relativity effects in their clocks. One finds that time itself goes a tiny bit slower on board the moving satellite than it goes for us on the ground, and this causes the onboard clock to disagree with the clocks on the ground. So we actually have to change our calculations to compensate for that or else the calculated location on the surface of the Earth is off. There's nothing wrong with the clocks, it's just they are so accurate that they can measure effects we don't generally notice in our daily lives.

Now if we start using our imaginations a bit, and think of spaceships people might use in the future to travel to distant stars, the effects predicted by this theory of relativity get even stranger. To get to a star 5 lightyears away in your lifetime, you would have to move much faster than the GPS satellite, say at 99% the speed of light. At this faster speed, the time effect is much more noticable. If an astronaut were to hop in one of these spaceships and fly at 99% the speed of light for 5 years, when she returned home at the end of that time she'd find that over 35 years have passed for the people she left back on Earth! For her it seemed like time flowed normally, and for the people on Earth it seemed like time flowed normally, but different amounts of time passed for each. This happens to you on a tiny scale in daily life too, but the speeds we go in cars and planes are much too slow to notice the time difference. We can, however, measure the effect in cases like the GPS satellite mentioned above.
The theory of relativity also tells us that time flows differently in strong gravitational fields too, like those very close to a star. We've measured this by watching the light from distant stars bend a little around our Sun. When we filter out the Sun's glare, we find the stars in the background are in slightly different places than when the Sun is not front of them. We've also noticed this gravity effect by looking at the orbit of the planet Mercury, the closest planet to our Sun. When measured very carefully, Mercury's orbit doesn't go exactly where Kepler's equations say it should go; instead it goes where the theory of relativity's equations say it should go.

In both the speed and the gravity examples described above, I haven't explained why we see those effects. That's where relativity gets a little more complicated, but it's really pretty easy to follow (until you hit the math of general relativity!). Let us for now say that distance, time, speed, and gravity are all far more intimately connected together scientists before Einstein thought, and this intimate connection causes strange effects like those I described above to become noticeable when the speed or gravity become large. Here are a few resources explaining where these effects come from: One starting place is Einstein's own 1905 special relativity paper, "On the Electrodynamics of Moving Bodies" (in html or pdf), which is amazingly understandable and describes everything in terms of moving clocks and rods. (This is a scientific paper and so does have some math in it, but overall is much more readable by the layperson than many scientific papers.) There are also nice technical writeups at Wikipedia on special relativity and general relativity, and a very well done website on the subject for the layperson here. Finally, there's a fabulous podcast by PBS Nova with various famous physicists describing the significance of E=mc², really enjoyable by scientists and laypeople alike. (I didn't mention it above, but E=mc² is a direct consequence of special relativity - maybe I should come back and fill in some of that...).