The animals in Minecraft are referred to as Passive Mobs. All living creatures in Minecraft are called Mobs and they are either passive, neutral or hostile. In this Instructable, I'm going to talk about the passive and the neutral mobs that don't usually cause you harm.

*6/21/16 - Updated for Rabbits and Polar Bears

Step 1: Leading Animals

You can use leads with passive mobs Just use it like you would anything else while pointing at the animal. You can tow them around behind you or you can attach the lead to a fence post to make the animal stay in place.

Crafting:

4 Strings + 1 Slimball = 2 Leads

Step 2: Sheep

Facts about Sheep

Sheep provide you with Wool which can be obtained from killing the sheep or shearing it with Shears (if you kill a Sheep you will always get only 1 block of wool, whereas if you shear it you have the opportunity to get up to 3)
Killing a Baby Sheep gets you nothing
A Sheep can be dyed itself, or you can dye the wool you get from the sheep; once a sheep is dyed it will always be that color no matter how many times it has been shorn, but it can be redyed
Breed a Sheep with Wheat
When breeding sheep of different colors together you will either get one color of the other, or you will get a whole new color if the colors are compatible to making a new one
Wool is needed to make Beds

Crafting

3 Wood Planks + 3 Wool Blocks = 1 Bed

Crafting Dye

Poppy / Rose Bush / Red Tulip = Rose Red
Orange Tulip = Orange
Dandelion / Sunflower = Dandelion Yellow
Blue Orchid = Light Blue
Lilac / Allium = Magenta
Peony / Pink Tulips = Pink
Azure Bluet / Oxeye Daisy / White Tulip = Light Gray

Dye Combinations

Lapis Lazuli + Cactus Green = Cyan
Lapis Lazuli + Bone Meal = Light Blue
Lapis Lazuli + Bone Meal + 2 Rose Red = Magenta
Lapis Lazuli + Pink + Rose Red = Magenta
Rose Red + Lapis Lazuli = Purple
Rose Red + Bone Meal = Light Pink
Orange + Dandelion Yellow = Orange
Inc Sac + Bone Meal = Gray
Cactus Green + Bone Meal = Lime Green
Gray + Bone Meal = Light Gray
Purple + Pink = Magenta

Others Dyes

Cook Cactus in Furnace = Cactus Green
Ink Sac from Squid = Black
Bone Meal crafted from Bone = White
Cocoa Beans = Brown
Lapis Lazuli = Blue

Step 3: Pigs

Facts about Pigs

Pigs drop Raw Porkchop when killed (if killed by fire or lava you can get a Cooked Porkchop if the Porkchop isn't destroyed by the fire first)
Pigs can be Saddled, but a Saddle cannot be taken back off of it
Breed Pigs with Carrots
If a Pig is struck by lightning, it will turn into a Zombie Pigman

Riding Pigs

A Pig can be ridden when you put a saddle on it
In order to ride it, get on the saddled pig and lead it around with a Carrot on a Stick
The carrot will eventually be eaten by the Pig

Step 4: Cows and Mooshrooms

Facts about Cows

Cows drop Raw Beef and Leather when killed
Can be milked over and over again with an Iron Bucket to get Milk - Milk can be consumed as an antidote for poison or used be an ingredient in cooking
Breed with Wheat
Baby Cows will not drop anything when killed, but can provide Milk

Facts about Mooshrooms

Only found in the Mooshroom Biome
Look like a cow except for Color and the mushrooms growing on its back
Mooshrooms drop Raw Beef and Leather when killed
Can be milked over and over again with an Iron Bucket to get Milk
Can be milked over and over again with a Wooden Bowl to get Mushroom Soup
Breed with Wheat
Shear a Mooshroom with Shears to get Red Mushrooms (once sheared it will turn into a normal cow and cannot change back)
Baby Mooshrooms will not drop anything when killed, but can provide Milk and be Sheared for Mushrooms

Step 5: Chickens

Facts about Chickens

Chickens can drop Raw Chicken and Feathers when killed
Chickens lay Eggs which can be used for recipes or you can throw and break them with the chance you will get a baby chick
Breed with Wheat, Melon, or Pumpkin Seeds
Unlike other animals, when they fall of a cliff, they won't take damage or die because they will flap their wings which slows their dissent

Step 6: Rabbits/Bunnies

Facts about Rabbits

Rabbits can drop Raw Rabbit, Cooked Rabbit, and Rabbit Hide. Also rarely drops Rabbit's Foot.
Use Cooked Rabbit to make Rabbit Stew.
Rabbit's Foot used for brewing Mundane Potion and Potion of Leaping
Killer Bunnies exist, but do not spawn naturally; they need to be spawned using commands - Have white fur with red eyes.
Breed with Carrots, Golden Carrots, and Dandelions.

Step 7: Horses, Donkeys and Mules

Facts about Horses

Horses only spawn in the Plains or Savannah
Do not spawn very commonly
Drops Leather when killed
Once a Horse is Tamed, it you can Breed it, Equip with Saddle and Armor (both a Saddle and Horse Armor cannot be made, they must be found) and Ride it
To ride on it and control it, the Horse must be equipped with a Saddle
Breed with Golden Carrot or Golden Apple
Unlike other tamed animals, a Horse will not follow you once tamed
Babies of tamed Horses will still need to be tamed when grown
Baby Horses can be fed Golden Apples, Golden Carrots, Apples, Wheat, Hay Bale, Bread and Sugar to speed up growing
Jumping while riding a saddled Horse, causes the Horse to jump
While riding a Horse you can jump over fences and stone walls

Facts about Donkeys

Only spawn in the Plains and Savannah
Spawn less often then Horses
Drops Leather when killed
Breed with Golden Carrot or Golden Apple
Needs to be tamed to equip it with anything; you repeatedly try to ride it until you get hearts to tame
Cannot be equipped with Armor, but can be equipped with a Saddle and Chest once tamed (provides 15 inventory slots) (chest cannot be removed once equipped unless the animal dies, then it will be dropped)
There is no variations in Donkeys, they all look the same

Facts about Mules

Created by breeding a Horse with a Donkey, does not appear naturally in the game
Cannot breed
Drops Leather when killed
Cannot be equipped with Armor, but can be equipped with a Saddle and Chest once tamed (provides 15 inventory slots) (chest cannot be removed once equipped unless the animal dies, then it will be dropped)
All Mules look the same

Taming Horses/Donkeys/Mules

To tame a Horse, Donkey or Mule try to ride the animal
The animal will most likely throw you off a few times
Once you see hearts, the animal has been trained, you can sit on it, but you can't ride it/control it until it has been saddled

Step 8: Ocelots/Cats

Facts about Ocelots

Are found naturally as Ocelots and become one of three breeds of Cats once tamed
Does not drop anything when killed
Found in Jungle Biomes
Will continue to follow a player and meow once tamed
Creepers are afraid of cats and will run away if one is nearby - makes them handy to have following you around
Are scared easily so are difficult to tame
Tame with raw fish
Breed with Raw Fish
If you can catch an Ocelot on a lead you can stake it to the ground and wait nearby for it to beg for food and you can tame it
A cat can be ordered to sit and stay instead of follow you

Taming Ocelots

First, find an Ocelot in a jungle
Have raw fish in hand
Find an Ocelot and try to get close enough without scaring away, then let it approach you
Will give off hearts and take on one of the cat skins once tamed

Step 9: Wolves/Dogs

Facts about Wolfs

Are considered Neutral mobs as they can be both passive and hostile
Does not drop anything when killed
Can be tamed by feeding Bones
Can be bred with Various Meats or Rotten Flesh
When tamed, a Red Collar will appear around the wolfs neck (it can be dyed another color)
When a Wolf is Hostile, its tail will stick out straight and its eyes will be red - it will attack and kill sheep whenever it wants but will only attack a player if it is attacked first
Once a Wolf is tamed it will follow you around unless you tell it to sit/stay
A tamed wolf will attack whatever the player attacks (except Creepers)

Step 10: Squids

Facts about Squids

They will not attack you even if you attack them
Are only found in water
Will drop Ink Sacs when killed which can be used as Black Dye

Step 11: Bats

Facts about Bats

Bats are passive mobs
They will not attack you even if you attack/kill them
Does not drop anything when killed
Are found in Caverns
They don't do anything except fly around and squeak in the dark

Step 12: Polar Bears

Facts about Polar Bears

Cannot be bred (that I know of).
Can possibly drop a Raw Fish or Raw Salmon, but you are not guaranteed anything if you kill one.
Will attack you if you attack it or if you get to near its cub, but normally cubs are Passive and adults are Neutral.
Can be led around on a Lead.