It's not really appropriate to say "time slows down." Regardless of what speed you're traveling, 1 second is one second long. Don't forget, we're buzzing along at a pretty good clip here on Earth as Earth orbits the sun. So, our clocks are "running slower" compared to the clock of someone stationary in our solar system. (just a very very small amount slower, as we're no where near the speed of light, see f95toli's equation.) There is no universal frame of reference.
Probably the easiest way to think about his is (allegedly) the same way Einstein began thinking about this. What would happen if you were traveling in a car at 99% the speed of light? We already knew at that point that no matter which direction we were going on earth, the speed of light was always the same. Thus, if we measured the speed of light reaching the earth from some star that the Earth is heading away from, it was the same 6 months later when the earth was headed toward that star. (as we go around the sun)
The speed of light is irrelevant to how fast you're traveling. If you can accept that last sentence, then it causes some problems. So, what if you are traveling at .99 the speed of light and turn on the headlights? Well, then you'll see light traveling away from you at the speed of light. Ahhhh, then someone on earth watching you turn your headlights on would therefore see light move away from you at faster than the speed of light - old style of thought: the speeds are additive. That leads to a contradiction, because we'll see it at the speed of light as well. The thing that reconciled this problem was that the rate of time was able to differ, relative to each observer. But, each observer perceives one second as being one second long.