Summary
Hyper Combos are high damage and powerful moves that can only be performed if you have the correct amount of Hyper Meter available to you. Most Hyper Combos take only 1 Level of Meter, but most (but not all) characters have Level 3 Hyper Combos that can only be performed with 3 Levels of Meter built up. Just like Special Moves, these require controller motions to perform but require that you activate it by pressing all three of the main Attack Buttons.
Have Meter, Will Use
When you perform a Hyper Combo, the screen will go into "Hyper Combo Freeze" where an animated close-up of the character performing the Hyper Combo appears and the background changes to a special Hyper Combo background. Once the Hyper Combo Freeze ends, your character will perform their Hyper Combo.
Strategy Corner
One interesting thing about Marvel Vs. Capcom 3 is that, during the Hyper Combo Freeze period, you are allowed to input codes of moves and they will register in their entirety during the Hyper Combo Freeze. In other words, as soon as you see the Hyper Combo Freeze, you can input the code for your own Hyper Combo completely. Right when the Hyper Combo Freeze ends, your Hyper Combo will execute instantly! There is no need to time your codes at all. Just input the code at any point during the Hyper Combo Freeze and as soon as the freeze ends, your move will execute. This even works with more complex motions like Akuma's Shun Goku Satsu.
So if your opponent activates a Hyper Combo and you want to 
it with your own Level 3 Hyper Combo, for example, don't mash the code, don't try to time it perfectly so that you finish the code right when their Hyper Combo Freeze ends... just input the motion, sit back, and watch it kick in.
Because Hyper Meter builds up so quickly in Marvel Vs. Capcom 3 and because you can hold up to 5 Levels at a time, Hyper Combos play a huge role in the game compared to Super Combos from other Capcom games. They are so readily available that you can use them far more often than other games.
Hyper Combos have a ton of practical applications. Most of the time, they are the best way to end combos because they do the most damage. Thus, Hyper Combos also do a lot more Chip Damage, so it can be used to finish opponents off with little life left. And lastly, the most practical application of Hyper Combos may be their ability to lead into a Delayed Hyper Combos (see Delayed Hyper Combo section for more details).
I Am Invincible... Or Not
While Hyper Combos are very powerful and do a lot of damage, many Hyper Combos have one main weakness: they have very little invincibility at start-up. As a result, they have very little ability to blow through the attacks of your opponent. Oftentimes, if your opponent's attack is inside you when you start a Hyper Combo, right when the Hyper Combo Freeze ends, your Hyper Combo doesn't even get a chance to activate before you get hit so nothing comes out at all.
Also, some Hyper Combos have a lengthy amount of start up time compared to other Hyper Combos. Storm's Hail Storm and Spider-Man's Ultimate Web Throw are two examples of Hyper Combos with longer than normal startup time. During that short period of startup time, the opponent can actually still move and, worse yet, they can actually hit you during these startup portions of the Hyper Combos. And even though you don't actually ever get to the point where your character's close-up animation plays, you still lose that Level of Hyper Meter from activating it.
Of course there are some Hyper Combos that do possess some startup invincibility. You can use these to overcome attacks from your opponent. Akuma's Messatsu Shoryu, for example, has a small amount of startup invulnerability, so if the opponent rushes at you and you activate that Hyper Combo, you can overcome their attack. You'll have to experiment with your character's Hyper Combos to figure out which ones have startup invincibility.
Utility Hypers
Some Hyper Combos activate a temporary power-up mode for your character or cause some other kind of lingering effect. In many cases, these "utility" hypers will persist on tag-out (allowing you to use powered-up assists as well as retain the power-up mode when that character is tagged back in) at the cost of not being able to build more meter with that character until the power-up has expired. Sometimes the reverse is true (can build meter while active, but effect vanishes on tag-out), and sometimes it's a mix of both.
Hypers that allow you to keep building meter but do not persist on tag-out:
Astral Vision, Kitty's Helper, Silent Kill, Healing Field, Chaos Dimension
Hypers that persist on tag-out but prevent meter building while active:
Devil Trigger, Vale of Mist, Sougenmu, Rimoukon, Berserker Charge
Hypers that allow you to keep building meter and persist on tag-out:
Viewtiful God Hand, Golden Armor, Mad Beast