What type of dancing do you throw your partner in the air?

You can do that in a variety of dances. Most commonly in Lindy Hop, but you do see it East Coast Swing as well. You see it occasionally in Salsa.

At any rate it is most likely to happen in a Showcase performance.

Maybe you should provide a little more information about the type of dance you are talking about. There are many types of dancing where one of the dancing partners is thrown in the air. In ballet for example there is a lot of lifting, jumping, and turning in the air.

It could be an exhibition swing dance like Lindy Hop.

At a social dance we don't do aerials, but most competitors do. In a social dance the floor is just too crowded for that stuff.

http://www.youtube.com/watch?v=mTg5V2oA_hY

I've seen some Salsa performances that the girl gets lifted. But that's only after a lot of practice and rehearsal and choreography.

I've seen it in Ballet and Lyrical.

Javelin practice?

the dangerous kind