Romeo is in a disguise because he and his friends attended a Capulet's party when Romeo is a Montague. And as everyone knows, Montagues and Capulets have a very bad relationship. So if the Capulets found out that one of their enemy came to their party, they would have started fighting, like always.

Romeo could have worn anything, as long as it really just covered most of his face.

This important to the drama because it shows that Romeo and Juliet really fell in love with each other at first sight, even with them not being able to see most of their faces. 

The most obvious answer is that he is a Montague, the son of the Capulets enemy.  He would not want to be recognized. He would wear a mask because it was a masked ball.  His goal was to see Rosaline, the current love of his life.

It was customary for the times for young men to crash parties.  If they were discovered, they simply had to dance for the guests.  Lord Capulet mentions that he remembers the times that he was young enough to participate in this sporty adventure.