Summary
To perform a Super Jump, simply tap one of the three Down positions on the controller for a fraction of a second before hitting any of the three Up directions. Remember: the Down position must be tapped before jumping. If you are holding Down on the controller and immediately shift to Up, you'll just perform a regular Jump. Down must only be pressed for a fraction of a second.
The Power of Super Jumping
The Super Jump is what has set the Marvel Series apart from every other Fighting Game series in existence. In most Fighting Games, you are only allowed to jump as high as one screen's worth of height. In other games where you are allowed to jump higher, it usually ends up being an alternate form of movement or just another means by which to try and approach your enemy. In other words, it's just an option.
In Marvel Vs. Capcom 3, Super Jumping is a way of life. In fact, you may spend a good portion of your game time Super Jumping around. Depending on your characters, it wouldn't shock me at all to have a character that spends more than 50% of their time Super Jumping into the air.
And why is that? It's because Super Jumps contain a lot of natural advantages. While on the ground, you have two different directions you must Block to properly defend all attacks. Blocking the wrong way will cause you to start getting hit. And while you can do that in a regular Jump as well, you are committed to the direction you are jumping. For example, a Forward Jump will have you moving forward with that Jump the entire time you are in the air.
Super Jumping, first of all, removes any issues of directional blocking: 100% of every attack can be blocked in the same direction during a Super Jump, which makes for a much tighter defense that is tougher to crack.
Super Jumping also allows you to guide your character in the air ever so slightly after they've started. The initial trajectory they go into is different depending on which of the three up directions you press on the controller, but once in the air, you can alter the trajectory ever so subtly. If you Super Jump forward, you'll never be able to start drifting backwards. You can just slow your forward momentum until down it stops. If you Super Jump straight up, you can glide left or right a maximum distance that's a bit father than a regular Jump's distance, but you start gliding very slowly and pick up speed more as you are coming down. If you Super Jump backwards, you can eventually get to a point where you are moving forward by the time you land.
Lastly, and most obviously, Super Jumping makes you jump at over twice the height of a regular Jump which means, if your opponent is still grounded, you have a distinct positional advantage of being at higher ground. Characters like Doctor Doom can make a living off of Super Jumping and raining Photon Shots down on their opponent. Not only that, but the viewpoint of the screen follows you up, and your grounded opponent will scroll off the bottom of the screen, represented only by a bubble that reads "P1" or "P2" with an arrow pointing to where they are.
Super Jumping Weaknesses
The main weakness of a Super Jump is that you cannot call Assists while in a Super Jump. This means once you take to the air for a Super Jump, you are on your own until you land. What most players tend to do, if they want to cover themselves, is to call an Assist before they Super Jump.
However, there are a couple of extra weaknesses that Super Jump possesses that make them not nearly as effective as a means of movement as they were in MvC2. The first thing is that Super Jumping isn't nearly as controllable as it was in MvC2. While you can alter your trajectory, the degree in which you can alter it isn't as dramatic as in MvC2. It's not a huge difference, but a noticeable one for MvC2 veterans.
The other change, however, is perhaps much more significant to the decrease in effectiveness of Super Jumping: your character doesn't turn around in the air, anymore, when the opponent goes under you. In MvC2, whenever the opponent passed under you, your character would automatically turn around if they were in mid jump. This allowed you to not only have a better idea of where the opponent was, but it gave your attacks while in the air a much better chance of beating the opponent on the ground. If, for example, if you were Cable and Super Jumped and the opponent ran under you, hitting Hard Kick would allow you to still perform your jumping attack facing the correct direction, so it could easily out-prioritize your opponent's move from the ground. In Marvel Vs. Capcom 3, however, your character will not turn around so, oftentimes, if the opponent runs under you, you need to find a good move that can cross-up as well to hit the opponent behind you, such as Haggar's Jumping Down + Hard Attack (the PIPE!) or Amaterasu's Glaive Mode Jumping Hard Attack.
Strategy Corner
The ability to break the defense of a character on the ground is a lot easier because of the need to block 
or low. But if your opponent just Super Jumps all day, how are you going to defeat a character that does not need to worry about blocking or low?
The answer is to make them have to guess blocking left or right. Because of the fact that the camera follows the Super Jumper into the sky and the grounded character scrolls off the screen, sometimes it becomes hard for players to realize which side you are on in relation to their character in the air. And, even if it's obvious that you are on one side and not the other, it's pretty trivial to dash under the opponent and get to the other side quickly and catch them by surprise from the other side. And given the fact that the opponent no longer turns around when you run under them, that gives you another advantage once you get behind them, as it is now harder for them to hit you. So a lot of the times when an opponent Super Jumps into the air, it's wiser to stay on the ground and try to cross under them to trip up their defense than it is to go into the air after them and try to defeat them up there somehow.
Also, remember that as soon as a character Air Blocks a move, they drop back down to the ground. So another tactic is to try and preemptively block the opponent from getting into the air with an attack that hits upwards or to hit them out of the air with such a move. The mobility of a character in Super Jump is still not great, so they are a pretty predictable target. Even if they can block it safely without worrying about having to block or low, the Air Block forces them to start dropping down. Once they get back down to the ground, quickly start your offense and do not let them back up into the air. Now you can pounce on them and get them worrying about blocking or 
again.
M.O.D.O.K.
A side note: although M.O.D.O.K. does not possess the ability to Jump, his Super Jump behaves like everyone else.