Super Mario Bros. 3 introduced a whole new whack of enemies, along with bringing back classics like Goombas and Koopa Troopas.
SMB3 introduced us to such enemies as Thwomps and Boos.

Angry Sun

The Angry Sun shows up twice in the game, first in World 2, and again in World 8.
It will swoop down from the sky and try to attack you. It can be defeated with a Koopa Shell.
The Angry Sun was a little scary as a kid.

Baby Cheep-Cheep

These little guys will come shooting out of a Big Cheep-Cheeps mouth, watch out.

Big Cheep-Cheep

This un-named Fish will show up in water levels, even levels that have land and water.
Go near him and he will eat you!
He will even jump out of the water very quickly to get you. Keep an eye open for this enemy.

Big Goomba

These are just like regular Goombas, but BIGGER! They appear in World 4.

Big Koopa: Green

These are like regular Green Koopa Troopas, but BIGGER! They appear in World 4.

Big Koopa: Red

These are like regular Red Koopa Troopas, but BIGGER! They appear in World 4.

Big Paratroopa: Green

These are like regular Green Paratroopas, but BIGGER! They appear in World 4.

Big Piranha Plant

These are just like regular Piranha Plants, but BIGGER! They appear in World 4.

Blooper

Blooper will show up in Water Levels. He will sometimes appear with mini-versions of himself, which he will shoot in 4 directions, trying to hit you.

Bob-Omb

Bob-omb's show up late in the game, they will walk toward you and blow up.
You can jump on them, and throw them.

Boo Diddly

Boo's appear in mini-fortresses.
They will come at you when you're not looking at them, and stop when you look toward them.
SMB3 introduced Boos to the Mario Universe.

Boom-Boom
    
Boom-Boom is in every world. He appears in the mini-fortresses. He can move very quickly.
He can jump very high, and occasionally has the ability to sprout wings and fly!

Boomerang Bro.

Boomerang Bros. will appear in a variety of levels. Like Hammer Bros. they will jump up and down, move left and right quickly. They will throw be throwing their boomerangs at you, and you must be careful, 'cause they're boomerangs, they'll come back at you from behind.

Bullets

Bullets appear in Flying Ships, as well as randomly in some levels throughout the game. These bullets will come shooting out at you. You can jump on them to destroy them, doing this will also give you a boost.
There are even Special Bullets in the game, which will turn around to catch you!

Buzzy Beetle

Buzzy Beetles will usually show up in underground levels.
They have a hard shell that you can jump on, then push them into other enemies.
Unlike Koopa Shells, they cannot be picked up.

Cheep-Cheeps
         
Cheep-Cheeps come in a variety of colours, and will appear mostly in water levels.
They will swim around, back and fourth, or just forward.
You can either avoid them, or shoot them with a fireball.

Chomp

Chain Chomps will be attached to a chain. But that won't stop them from trying to get you.
They will move quickly toward you when you go nearby.

Dry Bones

Dry Bones will appear in Mini-Fortresses.
They cannot be killed with a fireball, or by swinging your tail.
You can however jump on them, they will crumble into multiple bone fragments for a few seconds, then re-appear and keep walking. Creepy.

Fire Bro.

Like a Hammer Bro. but they will shoot fireballs at you.

Fire Chomp

Fire Chomps have a fire chain-tail, but that doesn't restrict them.
They will fly around shooting fire at you. You can jump on them to destroy them.

Firesnake

These guys appear in World 2. They jump (fairly widely) at you, try to avoid them.

Goomba

The classic Mario enemy. Goombas will show up all the time.
They will casually walk toward you.
You can simply jump on them, shoot them with a fireball, or just jump over them.

Goomba-In-A-Shoe

These are found only in level 5-3. The shoe gives the Goomba the ability to hop around.
You want to attack these enemies from below, by hitting the brick they are on.
This way you don't destroy this wonderful shoe.

Grounders

I've decided to call these mole-like enemies "Grounders".
They'll show up in underground levels, but also some above ground levels as well.
 They can move fairly quickly, and can pick up bricks and through them at you.

Hammer Bro.

Hammer Bros. will show up through-out the game. They even get their own mini-levels!
They will jump up and down, and throw hammers at you. Pesky little guys.

Hot Foot

Hot Foot will appear in Mini-Fortresses.
They will simply walk toward you. Similar to Boos, they will stop if you are facing their direction, and only move toward you if you are not looking at them.

Jelectro

Jelectro's appear in water levels.
They are stationary, but be careful, touch them and die!

Koopa Troopa: Green

Koopa Troopas are a classic Mario enemy. Green Koopas will simply walk forward.
Jump over or on them. In the first SMB, Mario could kick them when they were in their shell, in SMB3 you can pick the shell up, which adds a fair bit to the gameplay.

Koopa Troopa: Red

Red Koopa Troopas will walk back and fourth.
Like Green Koopa Troopas, once they are completely in their shell, you can pick them up.

Lakitu

Lakitu will fly across the top of levels and drop spiny eggs on you.
Lakitu tosses two kinds of eggs. One is a red egg, when it touches the ground will turn into a Spiny. The other is a green egg, which when it touches the ground will simply roll around.
Touching either will harm Mario.
Lakitu even show up in water levels, so watch out!

Lava Lotus

Lava Lotuses appear in some water levels.
They are stationary. They will open up and shoot at you, so you must avoid them.

Micro-Goomba

These little guys will come along with Para-Goombas.
If they land on you, you must jump quickly and repeatedly to get them off of you.
Micro-Goombas occasionally hide in a brick, and jump at Mario. You can see this quite a few times in the first level of World 2.

Muncher

Munchers can't move from where they are, but they can open and close their mouth and try and get you if you land on them. Level 7-7 is dedicated to them, it's also a very fun level.

Parabeetle

A rare enemy to encounter. These flying beetle's fly across the screen.
You can jump on them, this will not kill them or you.
You can hop from one to another to get further in the level.

Paragoomba: Brown

It's not only Koopas anymore who can get wings.
Para-Goomba's will show up throughout the game, they float around above you, and swoop in
to attack. Brown Paragoombas will drop little Micro-Goomba's on you.

Paragoomba: Red

Red Paragoombas will swoop down at you. They usually begin by walking on the ground, but they can hop and jump right at you!

Paratroopa: Green

Green Paratroopas will move horizontally. Sometimes flying, sometimes hopping up and down.
Sometimes back and forth, and sometimes just forward until they leave the screen or have something to bump into and turn around.
Jumping on them will turn them into regular Green Koopa Troopas.

Paratroopa: Red

Red Paratroopas will fly vertically, back and forth.
Jumping on them will turn them into regular Red Koopa Troopas.

Pipe Plants
    
These unnamed plants will come out of various pipes. Touch them and die.

Podoboo

Another returning enemy from the first SMB.
Podoboo appears in Mini-Fortresses and Bowser's Castle.
They jump out of lava. Gotta time your jumps to avoid them.

Ptooie

Ptooie shows up a few times in the game. They can walk around horizontally.
They also have a ball they shoot up and down.
You'll have to time your jumps to get past them, or just shoot them with a fireball.

Rocky Wrench

Rocky Wrench will appear on Flying Ships.
They pop-up from the ground and throw wrenches at you.
You can temporarily get rid of them bu jumping on them, get ready to play a little Wack-A-Mole.

Rotodisc

These appear in Mini-Fortresses and Bowser's Castle.
They repeatedly go 360 degree around a red ball.

Sledge Bro.

Hammer Bros' big brother. Like Hammer Bros they throw hammers at you. But unlike them, after they jump and land, the ground shakes, and if you're standing on it you will be unable to move for a couple seconds, leaving you open for a hammer attack. So watch out around these bros.

Spike

Spike will run toward you, if that's not bad enough, they will throw spiked balls at you.
Sometimes there are a few together throwing spiked balls at you, that can be annoying.

Spiny's Eggs
    
Lakitu will toss these at you from above. The green ones will remain an egg when they touch the ground, and roll around. The other will turn into a Spiny.

Spiny

These will form when a red Spiny Egg touches the ground.
They will also appear already formed in some levels.
Watch the spikes, if you jump on them you will die.

Statue

These statues will appear in Bowser's Castle at the end of the game.
Some are just regular statues that do nothing, but some will shoot a laser beam at you!

Stretch

This enemy will appear in mini-fortresses. It will move around its bar, it can even move below!

Thwomp

Thwomp's appear in Mini-Fortresses and Bowser's Castle.
They will come crashing down when you go near them, so watch out.
They aren't always positioned to vertically, they can also move horizontally, and even diagonally!

Venus Fire Trap
    
Venus Fire Traps will show up all the time throughout the game.
They live in pipes, and will come out of them to shoot fireballs at you.
They can shoot the fireballs up and down, left and right.

Walking Piranha
    
These little buggers will appear in a few levels.
They can jump up and try to bite you, and some can even walk. . .hence the name.