Ballroom shoes have suede soles, which provide the right amount of friction on the floor for dancing. Typically, shoes for latin dances are open-toed, and closed toe for standard and smooth dances (waltz, foxtrot, etc.) There are also practice shoes, which are usually lace-up with lower heels, easier on your feet than higher heels.
backbaydancewear.com has a huge selection of shoes, so you can see what the possibilities are. If you're looking for a versatile practice shoe, I'm partial to the low-heeled Godiva shoe available at dancehappy.com--it's much prettier than most practice shoes.
But you might want to ask your instructor about what kind of shoes you should start with. It can depend on what your dance goals are and how strong your feet are at this point, how used you are to wearing heels, etc.
One piece of advice you may get at a dance shoe store is that dance shoes should be very tight. I disagree with this and think it causes a lot of foot problems. Dance shoes should be snug enough that your foot doesn't slide inside the shoe, but they should not squeeze your foot into an unnatural shape or cause any pain.
