Summary
These are the basic attacks you perform when hitting any of the three main Attack Buttons. Obviously, the strength of the button usually determines the speed and strength and range of the moves: Light Attack causes faster but low damaging and short-ranged attacks. Hard Attack causes high damage moves that usually reach farther, but they are generally much slower. And Medium Attack will perform moves that, not shockingly, fall in between. Of course, this varies greatly from character to character, particularly in a game like Marvel Vs. Capcom 3 where the characters are so unique and varied. But for the most part, those concepts hold true.
Types of Normal Attacks
There are three main states of Normal Moves, and learning your character will take knowing all four intimately. The three states are Standing (hitting an Attack Button while your character is standing, Crouching (hitting an Attack Button while you are also holding any of the Down positions on the controller), and Jumping (pressing an Attack Button when in the air). It is also noteworthy that, while jumping, the Special Button gives you a fourth button of Normal Attacks. On the ground, the Special Button only performs Launchers, so they do not count as a Normal Attack.
On top of those three states, many characters have Unique Attacks, which are Normal Moves that can only be accessed by holding the controller in a certain direction. For example, Ryu has a move that can only be accessed by holding Toward on the controller and hitting Medium Attack: his Overhead Punch that cannot be crouch blocked. Wolverine has a sliding attack along the ground that has to be performed by holding Offensive Crouch and hitting Medium Attack. Doctor Doom can perform the Hidden Missile attack by holding Back on the controller and hitting Hard Attack repeatedly. She-Hulk has a move while jumping that can only be accessed by hitting Down on the controller and pressing Hard Attack: her Reverse Splash.
Almost every character has a Unique Attack, so make sure you know what they are for your favorite character. Unique Attacks usually have unusual properties associated with them (e.g. they are Overheads, they can perform Chip Damage, they cause Soft Knock-Downs, they reach farther than every other Normal Attack, etc.), so it's a good idea to get to know them well.
Strategy Corner
Really take the time to get a strong grasp on all the different move properties that exist in the game. Refer to the glossary page here (Attack Properties) and understand what each property means. Once you get a strong grasp of these properties, memorize which of your moves has which property. Once you learn all of that, that's when you can really start taking advantage of all your moves.
Starting with Normals
One of the main benefits of Normal Attacks are their ability to be canceled into another Normal Move (called Chaining) or a Special Move or a Hyper Combo (called Buffering). Particularly with Buffering, you can really start using a proper combination of Normal Attacks and Special Attacks to really do some damage in combos or perform some really intense close quarters rushdown on your opponent or allow you to keep opponents at a distance a lot easier.