The Binding of Isaac Rebirth introduced a couple of transformations to the game, but the Afterbirth DLC comes packed with an additional 9 to play with and Afterbirth+ introduces 3 more.
A transformation gives you special abilities in addition to your current form and each one holds a unique effect to help you along your way. The criteria for each one is listed below and they all involve picking up a specific combination of items.
Isaac turns into his cat Guppy and gains the ability to fly. Every time you successfully hit with a tear, a blue fly will spawn (In Repentance this was nerfed to 50% of tears that hit). Taking any Three of the following items will trigger the Guppy transformation.
Isaac turns into a human fly and gains the ability to fly. This transformation will turn most types of flies into friendly blue attack flies. Taking any Three of the following items will trigger the Lord of the Flies transformation.
Isaac turns into his mother, giving him a knife which is attached to his butt and cannot be fired. Taking any Three items in this list will give Isaac the Mom transformation.
Isaac turns into a tentacled monster, gaining 2 black hearts and the ability to fly. Taking any Three items in this list will give Isaac the Leviathan transformation.
Isaac turns into a faceless angel, gaining 3 soul hearts and permanent flight. Taking any Three items in this list will give Isaac the Seraphim transformation.
Isaac turns into Bob (Sloth) and will now leave a trail of green poisonous creep as he walks. Taking any Three items in this list will give Isaac the Bob transformation.
Isaac turns into a living Mushroom and gains +1 HP. Taking any Three items in this list will give Isaac the Mushroom transformation.
Isaac grows two tumors on his face, which will fire diagonally in addition to Isaac's normal tears. This transformation is also a -0.3 damage down and -0.3 tears down.
Taking any Three items in this list will give Isaac the Conjoined transformation.
Isaac turns into a buff berserker gaining +2 Damage, +0.15 Speed and spawns a random pill on the floor. Taking any Three items in this list will give Isaac the Drugs transformation.
Isaac turns into a giant walking turd and he can now heal half a red heart each time a pile of poop is destroyed. Taking any Three items in this list will give Isaac the Poop transformation.
Picking up all 3 bum familiars will combine them and transform into SUPER BUM. Super Bum will pick up any of the other bum's pickups and drop double rewards. The Three items required are:
AFTERBIRTH+ ONLY
Isaac gains a monocle and sometimes shoots 2 shots at once, similar to the 20/20 effect. Picking up any Three book items in this list will give Isaac the bookworm transformation:
AFTERBIRTH+ ONLY
Isaac gains blue friendly attack spiders every time he takes damage and gains a permanent familiar spiderbro, which applies a random status effect to any enemies that touch it. Picking up any Three spider-related items in this list will give Isaac the spider baby transformation:
AFTERBIRTH+ ONLY
Isaac grows hair all over his face and gains +1 HP. Taking three puberty pills will give Isaac the adulthood transformation:
AFTERBIRTH+ ONLY
Isaac creates dust clouds when walking and has a chance to create rock waves around him after taking damage, which will spread outwards and damage enemies. (In Repentance this was buffed, you also can walk over rocks and destroy them) Taking three items or pills that increase Isaac's size will give Isaac the Stompy transformation.