Just as there is no single protagonist, there is no single antagonist in this wonderful play. There are several protagonists, grouped together (the "rude mechanicals," the lovers, etc.), and several antagonists.
If forced to pick a single figure, however, I'd say Puck fills the antagonist role most emphatically. He complicates Oberon's plans. He is the engine of transformation who sets Bottom against his friends (and vice versa) and the entire Titania love theme in play. He tricks and misleads the lovers, so they are fighting him when they think they are fighting one another.
Well this question really depends on the part of the play. If you're taking things from the point of view of Lysander and Hermia, it could be Demetrius or Hermia's father. Both attempt to stand in the way of the arranged marriage between Hermia and Demetrius... Demetrius out of love, and her father out of scorn for Hermia's refusal to follow his instructions. In fact, he even goes as far as to invoke the law of Athens, which allows a daughter who is disobedient to be killed for not listening to her father.
Also, you could view Puck and Oberon as antagonists, as they mess around with the young Athenian lovers' feelings. By charming both Demetrius and Lysander, the former on purpose, and the latter by mistake, they believe they are helping Helena to find love. However, in reality, they are simply ruining things and causing more problems than they are good, regardless of their intentions.