Summary
After being struck by a move that causes a Hard Knock-Down or a Soft Knock-Down, hold the controller Toward or Back to roll in that direction. When struck by a move that causes a Forced Tech Roll, you cannot perform a roll at all.
Re-Positioning Yourself
First things first: Rolling in Marvel Vs. Capcom 3 will not allow you to escape any combos. This is stated right away because, in MvC2, Rolling was a technique that allowed you to avoid combos that could hit you after you became grounded. In this game, Rolling is simply a means by which to re-position yourself when you get up off the ground. Rolling will never lessen the amount of time you are lying on the floor.
If you do not hold any directions after getting knocked down, your character, when they get back on their feet, will perform what is referred to as a "Tech Roll" after a similar move from previous Street Fighter games like Capcom Vs. SNK 2. The character rolls backwards a very short distance and immediately gets up onto their feet.
When you hold a direction, however, your character will roll a much longer distance in the direction you hold the controller and get up on their feet at the end of this roll. You are completely invulnerable during this roll, so there is no way the opponent can hit you out of it. The roll is, however, a fixed distance: there is no way to alter the distance your character travels.
Strategy Corner
Rolls have a fixed distance, so if you anticipate your opponent rolling, you can actually get to the area where their roll will stop and have a surprise or trap waiting for them. You can choose to lock them down the 
the roll ends or just go for a Throw or go for a high/low mix-up. For example, opponents rolling out of the corner becomes a very obvious tactic at some point, so it's easy to anticipate it and be ready to punish them for it. Even you cannot directly punish a roll, you can definitely set yourself up so that they can be caught off guard at the end of their roll if you anticipate it and meet them where they stop.
So it becomes a guessing game of sorts on wake up. Do you roll left or right or maybe just get up in place? Smart opponents can anticipate where you go if you roll the same direction every time, so learn to vary it up a bit, just like with Air Recoveries. The easiest way to lose fights is to start becoming predictable, so learn to vary up your roll directions.
Rolling With the Homies
One very important aspect about Rolling is that, during the roll, you are actually free to perform Cross-Over Assists! That means, right when you start to roll, you can press the Assist Button and, even before you finish rolling, your Assist, depending on how fast they are, can have already come out and started attacking. This is particularly useful if you choose to roll through your opponent and rise up on their other side. Your Assist will be called on the original side and you'll get up on the other, essentially sandwiching your opponent between you and your Assist.