RandFan wrote an excellent and moving post about his decision to go on a mission when he wasn't sure of his testimony. I'll have to go back and see if I can find it.
All young men are pretty much expected to go on a mission which they and their families finance. Young women may go if they choose, although they go at 21 while men go at 18. (That was the way it was when I was a kid, anyway. Women were expected to look towards marriage and children, so a mission isn't required for them.)
They are sent to the
Missionary Training Center in Provo, Ut for a few months where they learn everything they need, including being taught a new language if they're going on a foreign-language speaking mission. They go immediately into the field.
The missions themselves last 18 months or 2 years, and their interaction with their home and families are strictly curtailed (and really anything outside of their mission, and the church). It's a very difficult experience, but to my knowledge, most people consider it a genuine highlight of their lives.
ETA: Ninja'ed by RandFan who was much more eloquent and much more informative.