I have a theater / dance class and was told to bring dance shoes what would these qualify as?

Most likely ballet shoes
Payless sometimes has them.

Or character shoes... if it's a theatre class, they may want you to have heeled or non-heeled character shoes. Character Shoes - Dance 4 Less

If unable to purchase regular dance shoes then, just plain slippers might be okay to use.

When one specifies "dance shoes" one generally means soft pliable shoes with non-rubber soles/heals. No sneakers. No platforms. No heals. Something you can move easily in. Ballet slippers are OK - except if they are toe shoes.
Sometimes dance shoes are called "character shoes." They have a very shallow heal.
Check out Capeziodance.com and find a store near you.