I assume you are talking about Act 1, scene v, when Romeo attends the Capulets party. Remember that Romeo is esentially in a no man's land, as he is in the house of his sworn enemy, an enemy with whom he had an altercation earlier in the day before the party. It was absolutely necessary for him to wear a disguise for the event, and for the period, hand-held masks, for both men and women were not uncommon. This allowed him the freedom to reveal himself to Juliet at their meeting, but to keep it hidden when necessary, as with Tybalt, although he discovered his identity anyway in the long run. It is also important to the drama in other ways, as well. It serves to heighten the animosity Tybalt feels for Romeo and all the Montagues because he has had the nerve to appear in the Capulet house, even in disguise, and Tybalt feels as if the Capulets are being mocked by the Montagues.
Romeo must go to the Capulets' party in disguise because he is a Montague, and the two families are feuding, perhaps even warring. If he weren't in disguise, he wouldn't get in. Therefore, it is somewhat important to the scene's tension. It's also important because it allows both he and Juliet to fall for one another without knowing who the other is.
As for what he could wear, it should be something relatively simple, so he can drop it when stunned by Juliet's beauty (and, practically, so Tybalt can pierce it easily).
Romeo had to go to the Capulet's party because his cousin and Mercutio told him to go and get over his old crush. He had to be in disguise because the Montague and the Capulet do not like each other, so if the Capulet found out their enemy came to their party a fight would start.
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.