They do a whole battery of tests, some on paper - multiple-choice questions to determine whether you have any form of psychological pathology. Some are verbal questions like word problems in math class. Some are puzzle-like tests in which you are asked to regroup blocks into patterns you've just seen. There's the old standby Rorshach inkblot test where you look at splotches of colored ink on paper and tell what it looks like to you There are free-association word tests, in which you hear a spoken word and say the first thing that comes to your mind when you hear it.
There's more, but I can't remember them all. It doesn't hurt and it wasn't scary or embarrassing, other than a little mild stress when I felt like I was supposed to know the 'right' answer and I wasn't sure what that was (I was a bit frustrated at the puzzle-pattern stuff, which I know I could have done better at, if the psychologist hadn't been WATCHING ME, but perhaps that was the point of the test). Mostly it was a very interesting experience for me, I think you'll find it fascinating. Takes a few hours (not necessarily all at once - it may be broken up into a couple of sessions on different days), then you'll probably be asked back later for an interview in which the psychologist will talk to you and your parents about the results of the testing and what he/she/they feel it says about you.
How it's done can vary quite a bit depending on the standards of procedure at different facilities and different doctors, but I think the testing is pretty much the same. From descriptions I've seen here when this question has come up before, everyone who goes through it comes away with a different perspective on what exactly happened, different aspects strike different personalities as more memorable than others. 