How do I ask for a dance?

Most often, you just walk up and ask someone. Usually a man will ask a woman to dance, but at some milongas, in some cities, it's acceptable for women to ask... and it's accepted at most venues for women to dance with women or men to dance with men (but very few in Buenos Aires).

However, there is a charming convention that originated in Argentina called the cabeceo (ka-ba-say-o) that involves making eye contact. If there's someone you want to dance with, you stare at them until they look your way. When you make eye contact, you raise your eyebrows and/or make a subtle head nod (cabeceo = nod) toward the dance floor. The responder answers "yes" by smiling and nodding back, or "no" by turning and looking away. (It's polite to wait until your potential partner has come off the dance floor before you ask them for the next dance.)
The key thing to make this work is that you must make eye contact. In American culture that can be considered aggressive, but, well, you just have to get over that if you want the cabeceo to work.