One of the things that made SMB3 such a hit, is Mario's ability to wear a variety of suits.
This has become a staple of Mario games, and can be seen in many games since.

Small Mario
    
This isn't exactly a suit, just how Mario begins the game. His classic overalls and red cap.

Big Mario
    
If Mario snags a mushroom he gets bigger. Standard stuff, we saw in the first SMB.

Fire Mario
    
Another suit which was brought back from the fist SMB.
Mario can shoot fireballs with the click of a button, which defeats most enemies.

Raccoon Mario
    
This is the suit the game is known for. This is the suit that graced the cover of SMB3.
For the first time, Mario could fly! And it was amazing.
Not only that, he could use his tail to attack enemies.

Frog Mario
    
I loved the Frog Suit as a child.
It allows you to gracefully swim in water levels, and adds a fun difficulty to non-water levels.
Try to beat a Koopa Kid with one of these.

Tanooki Mario
    
This took the Raccoon Suit to a whole new level. Not only can you fly and use your tail to attack, but you can turn into an invincible statue for a few seconds.
You are also able to kill rotodiscs while in this statue form, quite cool.

Hammer Bro. Mario
    
Such a cool and rare suit. When you are able to get one of these suits, you can throw hammers!
Just like those pesky Hammer Bro enemies.
Fight Hammers with Hammers!

P-Wing(Paratroopa Wing)
    
This suit looks and performs exactly like the Raccoon Suit in levels.
Except for one thing. . . . . .IT HAS POWER!!!
You don't have to run and get speed to fly, nor does your flight run out.
You can fly and fly until the time runs out, or you hit an enemy.
Once you finish the level, the P-Wing's power is gone, and it's a standard raccoon suit.
Once you defeat Bowser and restart the game in World 1, you are rewarded with your item box filled with P-Wings!

Kuribo's/Goomba's Shoe
                             
Kuribo's Shoe is a classic to SMB3 fans. This shoe only shows up in one level, 5-3.
A Goomba is originally wearing it, but if you can hit him with a brick from underneath, you got yourself a shoe!
This shoe allows you to walk on Munchers! Yes, you feel like a King (or Queen) wearing it.