1x is the speed that DVDs are read when they are played for video on a DVD player. This would involve a 60-minute burn time at that speed for a full disc because that's how long a full disc would play.
2x, 4x, 8x, etc. theoretically burn the disc at this faster speed. I say theoretically because the reality is that the higher burn speeds only happen at the outside of the disc. Few burners (if any yet) can burn the inside faster than 4x.
What this means is that although a 2x burner is twice as fast as a 1x and a 4x is twice as fast as a 2x (and takes about 15 minutes), an 8x takes 8 minutes (a little over half as long) and a 16x takes 6 (only 3/4 of the time an 8x takes).
Fast burners can burn at slower speeds and sometimes this is good. Some media won't accept burns at faster speeds and usually you get better quality at middle speeds.
My burners are 8x but I usually burn at 4x.