Summary
To perform a Throw, hold the controller Toward or Back and press Hard Attack to perform a Forward Throw or a Back Throw respectively. Forward Throws always put the opponent on the same side they started on and Back Throws always put the opponent behind you. You can only perform a Throw when near your opponent while both of you are on the ground.
All the same rules apply for an Air Throw with the only difference is that and your opponent must both be in the air to connect an Air Throw.
To perform a Break Away, simply hit Hard Attack while holding any of the three Toward directions or any of the three Back directions on the controller immediately after your opponent lands a Throw or Air Throw on you to escape their Throw.
Increased Role of Throwing
Throws didn't play a very large role in MvC2. In fact, you almost never saw them because landing a Throw was such a difficult task. In Marvel Vs. Capcom 3, however, Throws have been given quite a boost in effectiveness, and you can expect to see many more Throws throughout a game.
The reason Throws have become more powerful is that they have been given heftier ranges in Marvel Vs. Capcom 3. Also, they seem to come out quicker or have decent priority. What the exact factor is is not clear just yet, but regardless Throws seem to connect a lot more than before.
Everyone's Throw and Air Throw affords them different benefits. Some Throws drop the opponent onto the ground for a Hard Knock-Down that gives you plenty of time to combo off of it. Other Throws cause the opponent to recover instantly so the direct damage off the Throw is the only reward you get. But for those characters that can combo off a Throw, the reward can be huge. Characters like Sentinel can K.O. an opponent with half of their life left just by landing a Throw into the corner. It's definitely worth learning your best combo off of a Throw.
No Cross Throwing
You cannot Throw or Air Throw an opponent while they are in Hit Stun or Block Stun. And you cannot ground Throw an opponent who is in the air and cannot Air Throw an opponent on the ground. You cannot Throw opponents when they are floored and you can never chain into a Throw.
Break Aways
The only way to escape a Throw is to perform a Break Away. However, the window to input a Break Away is very, very small. In fact, you probably will almost never be able to react to a Throw and escape it via a Break Away. It will almost always require anticipation or, if you're lucky, even accidental Break Aways when you meant to actually attack the opponent with the Hard Attack and get thrown right beforehand.
Break Aways can also be performed in the air. And after an air Break Away, both characters fall away from each other and immediately start falling to the ground. However, either character can immediately perform any action on their way down.
Strategy Corner
You can actually defend from Throws pretty effectively if you combine trying a Break Away with trying an Advancing Guard. Holding Back or Down/Back and pressing Hard Attack will break a Throw, but so will holding those directions and pressing all three Attack Buttons, which happens to be the command for an Advancing Guard. So if the opponent comes up to you and you're not sure whether they are going to attack or Throw, you can press {{ 
and h}} at the same time to handle both situations: if the opponent attacks, you perform an Advancing Guard and if your opponent went for a Throw instead, you'll perform the Break Away. It will still require some good timing to be able to cover both situations with a single timing of pressing the three Attack Buttons, but it can be a valuable way to defend against both options at the same time.
Special Throws
Keep in mind that all of the above rules apply to regular Throws. Many characters have Throws that are Special Moves or Hyper Combos, called Command Throws, that have all sorts of different properties that may violate some of the rules that apply to regular Throws. Also, all Command Throws cannot be escaped via Break Away.