No registration is required, and you can compete against yourself or others by a race car that moves fast or slow depending on the speed of your typing. It shows your errors, and you can continue until you fix it.
I’ve placed the link on my website under Typing. You can find a lot of other typing games there, as well.
(Note: After I posted this, Ben Rimes pointed out in the comments that he found some inappropriate movie quotes being used for typing samples in TypeRacer. When I had tested it out, it just used appropriate quotes from books. So you might want to be careful using this in the classroom.)