Summary
Special Moves give every character in Marvel an arsenal of more powerful tools alongside the Normal Moves they have. Special Moves require a specific controller motion or a combination of buttons to perform. Examples of Special Moves are Captain America's Shield Slash (which requires a Down, Offensive Crouch, Towards controller motion before pressing Punch), Spider-Man's Web Zip (which requires you to hit the Special Button with any one other Attack Button simultaneously), and Chun Li's Hyakuretsu Kyaku's (which requires you to tap any of the three main Attack Buttons as fast as possible).
Chip Damage
The main thing that makes Special Moves significant besides the more complex motions are the fact that they do damage even if the attack is blocked. This is known as "Chip Damage" or "Block Damage". Chip Damage is always 30% of the actual damage. You can actually be K.O.'ed from Chip Damage. So even if you Block the Special Move, if you do not have enough life left, you will lose.
Strategy Corner
Chip Damage plays a huge role in Marvel Vs. Capcom 3. In fact, sometimes Chip Damage is more threatening that actual damage. So if you want to learn to play Marvel Vs. Capcom 3 correctly, you will have to start really implementing Chip Damage as part of your gameplan. Some players may not be able to properly grasp the idea that Chip Damage can be such an effective way to drain life off of the opponent. However, don't be frustrated if the opponent keeps blocking your Special Moves or Hyper Combos: keep them coming! In fact, some players will burn Hyper Combos specifically just to do 
Chip Damage -- they don't care at all if the opponent blocks it or not. All that Chip Damage adds up quickly in Marvel Vs. Capcom 3, and in some cases entire characters can lose the majority of their life through Chip Damage. This would not be a rare occurrence at all.
So definitely start getting used to the idea of planning on getting as much Chip Damage as possible if your character is equipped for it. Doctor Doom, for example, is particularly well equipped for causing tons of Chip Damage via Hidden Missiles, Plasma Beams, Photon Shocks, and Molecular Shields. It's definitely a very strong strategy to damage opponents via Chip and, in many cases, will unnerve them to the point where they make defensive mistakes and open themselves up for big damage. Yes, the combo is what did the actual damage, but it was the Chip Damage that caused them to make that mistake to get comboed.
Controlling Space
While Normal Moves will form the basis of your gameplay, Special Moves can actually be the majority of your offense and defense. Because Marvel Vs. Capcom 3 is a game that can be played at all ranges and Normal Moves generally don't reach that far, Special Moves play a very prominent role in locking your opponent down or keeping them away from you or attacking from a safe distance. They will be your main tools when it comes to controlling the space in the playing field. Remember, characters in this game have extreme mobility thanks to all the different ways a character can move, and learning how to use your Special Moves to control what areas your opponent has freedom to move into will be key to defeating your opponent.
Combos
Special Moves also generally form the basis of extending your combos. While most characters can do a basic Chain Combo into Launcher into Air Combo sequence, the key to making your combos more potent and damaging will be using Special Moves, as Special Moves, more often than Normal Moves, come with those special move properties such as Ground Bounce, Wall Bounce, Hard Knock-Downs, and such. And because of that, not only do they allow you to land longer combos, but they let you more easily combo into Hyper Combos, as Special Moves can be canceled into Hyper Combos.
Special Ingenuity
Mastering Marvel Vs. Capcom 3 will require you to really learn how to use the Special Moves in ways you may not have realized they were meant to be used. Thanks to the huge variety and quantity of Special Moves, the ability to use your imagination and ingenuity to get the most out of your Special Moves is almost limitless, so experiment as much as possible.