And yes, I'm going to use the movie Captain America: the First Avenger as an example because it has a strong midpoint scene. And because it's an awesome movie, of course.
At the midpoint of the story is usually a victory for your hero to get him pumped up and cocky right before the Bad Guys Close In.* In Captain America the midpoint is the part where Cap quits being a chorus girl and goes to rescue Bucky and the rest of the 107th from the clutches of Hydra.
He's never fought in a battle before so he's untested and we the audience don't really know if he's able to break into a Hydra facility and save his friends all by himself. When he succeeds, he proves himself to be a super hero capable of doing more than just punching a fake Adolph Hitler.
At your midpoint, give your hero a chance to prove himself both to the other characters in the book and to your readers. We want to know if this guy really has the potential to be a hero. Plus, since the second half of the story tends to get grittier as it goes, we need a moment of triumph to sustain us through the darker part of the journey.
K. M. Weiland @ Helping Writers Become Authors wrote an excellent post on how the midpoint moves your hero from action to reaction. She used Captain America: The Winter Soldier as an example, but the same thing happens at the midpoint of the first Captain America movie as well.
Up to the midpoint of the movie, Cap knows about Hydra and Johann Schmidt but he doesn't know the full extent of what they're up to (although the audience knows via a few well-timed villain lair scenes). When he breaks into Hydra's lab, he uncovers the sinister plot, thus catapulting the movie into its second phase: namely, taking out Hydra before it can destroy the world.
When your story reaches its midpoint, it's time for the hero to switch from discovery to action. Instead of letting the events of the story knock him around, the hero starts to get in the game and throw a few punches himself.
Sometimes the hero meets the villain right at the start of the story, but usually the villain remains shrouded in mystery for the first half of the book or so. In Captain America, Cap doesn't meet Schmidt until the midpoint. The two super soldiers know of each other's existence, but they've never met face to face until Cap destroys the Hydra plant, forcing Schmidt to flee.
At this point they have a brief interchange and Schmidt reveals himself as the Red Skull. They also get an opportunity to punch each other and test which of them is superior. Up to this point Cap's battle is against a faceless enemy, but when he meets Schmidt, the true enemy is unmasked (literally) and from there on out the battle is personal.
You'll probably want to save the epic battle between good and evil for the climax of your story, but throwing in a brief showdown at the midpoint lets the hero and villain probe each other's weak points and gives readers a taste of what's to come.