Relativity is best known for giving us a new view of time. The faster you go, the slower time moves. There are other effects as well. For example, the faster you go, the more squished you get.
The most famous consequence of Einstein's work with relativity is the idea that time slows down as an object speeds up. The less-famous fine print is that time seems to slow down only for the person who is watching the movement. If two twins synchronize their watches before one steps onto a train, the synchronized watches read the same right until the train starts moving. The twin that is on the train - let's call her Maria - won't see her watch slow down. It'll be right on time, from her perspective. The twin at the station, however, will check her watch with Maria's and see that Maria's watch is slowing down.
Who is right? Both of them. Researchers have done experiments that show that unstable particles decay more slowly if they are accelerated to high speeds. Time literally is slowing down for them. How can time slow down for both the particles and Maria - without them noticing it?
When you adjust time, everything else adjusts along with it, and that includes length. One of the consequences of relativity is known as "length contraction." How do we know this? Because it helps us solve a problem like Maria's. (I'm sorry.)
The one thing that the twins both agree on is the speed at which they are moving apart from each other. That should make it easy to figure out which one is wrong about the time. If the train is covering a yard a second, and they simultaneously measure after its distance from the station, they'll know whose watch was right. Except they don't. The twin at the station measures after ten seconds, and sees the train as ten yards away. But Maria's watch shows that the train has been moving only nine seconds. And when she measures, she sees the station as nine yards away. Not only does relativity warp time, it warps space.
The same thing happens for those decaying particles. If you calculated the time dilation for their speed, you could race them along a track, and put up a "finish" line where they would make their long-delayed decay. Watching them decay from the sidelines, you would see them move a long time at a high speed before they decayed, making them long-lived. If you were traveling along with them, checking your watch, you would see them decay at a normal rate. They'd decay at the finish line. But that finish line would not look far away. It would be close up. So you'd see them living a normal, non-protracted life, and make a short journey to a nearby finish line. With time dilation comes length contraction.