Do I need a costume to play?
Not at all. Only if you want to save $2 on admission! Be aware that if you wish to play and your character does not have a costume, one will be provided for you. Thus you may have no control on how you will be dressed, so it's probably a good idea to bring your own.

What constitutes a costume?
Something you would not normally wear in public without expecting some ridicule (Renfair does not count). Basically, something that suggests a character. You could have jeans, but wear a medieval tunic. A simple bodice is fine. All in all, it is up to the decision of the staff.