One of the new features added to The Binding of Isaac: Rebirth is local co-op multiplayer, where the second player controls a floating baby. It works in a drop-in/drop-out style of play where the second player steals a heart container from Isaac upon spawning. Damage is lowered to balance things out but every aspect of Isaac's items are copied over to the second player's baby.
Every time you spawn in a new baby it is randomly picked from one of the babies in the HUGE list below (There are 203 in total). Most of the baby skins are cosmetic, however quite a few also have their own special abilities.
Spawns friendly spiders
Drops 1-2 hearts when it dies
Spawns with a pretty fly
Explodes into tears when it dies
Fires homing tears
Sometimes spawn troll bombs
Re-arranges shooting keys after every floor
Shoots tears diagonally (Similar to R U A WIZARD pill)
Spawns non-friendly spiders on death
UNLOCK: Beat Mom's Heart with Isaac on Hard Mode to unlock this baby
UNLOCK: Beat Mom's Heart with Maggy on Hard Mode to unlock this baby
UNLOCK: Beat Mom's Heart with Eve on Hard Mode to unlock this baby
Higher damage tears
UNLOCK: Beat Mom's Heart with Judas on Hard Mode to unlock this baby
Drops random pickup on death
UNLOCK: Beat Mom's Heart with Cain on Hard Mode to unlock this baby
Drops coins on death
Fires a constant laser (Same as Tech 2)
Farts when hit
Has the magneto effect (Attracts pickups)
Spawns bomb pickups on death
UNLOCK: Beat Mom's Heart with Azazel on Hard Mode to unlock this baby
Fires fear tears
UNLOCK: Beat Mom's Heart with Lazarus on Hard Mode to unlock this baby
UNLOCK: Beat Mom's Heart with ??? on Hard Mode to unlock this baby
UNLOCK: Beat Mom's Heart with Samson on Hard Mode to unlock this baby
Immune to all damage
UNLOCK: Beat Mom's Heart with The Lost on Hard Mode to unlock this baby
Unable to shoot but can pickup consumables for Isaac
Reference to Edmund's game Time Fcuk
Reference to the game Alien Hominid
Reference to Commander Video from the Bit Trip series
Edmund McMillen baby (Creator of the Binding of Isaac)
Danielle Baby (Edmund's Wife)
Reference to Edmund's game Aether
Reference to Brownie from Super Meat Boy
Reference to Captain Viridian from VVVVVV
Unable to attack if the player picks up Mom's Knife
No special effect.
Reference to Pikachu from Pokemon
Reference to Aban Hawkins from 1001 Spikes
Reference to Bandage Girl from Super Meat Boy
Reference to default character in Spelunky
Reference to Dr. Fetus in Super Meat Boy
Enter the Gungeon