Forecastle: Where the crew's meals were cooked, and where the crew's food and supplies were kept.
Poop House:  Nothing to do with a bathroom, the poop house was the living quarters for the ship's master (Christopher Jones) and some of the higher ranking crew, perhaps master's mates John Clarke and Robert Coppin.
Cabin:  The general sleeping quarters for the Mayflower's twenty or thirty other crewmembers.  The crew slept in shifts.
Steerage Room: This is where the pilot steered the Mayflower.  Steering was done by a stick called a whip-staff that was moved back and forth to move the tiller, which in turn moves the rudder.
Gun Room:  This is where the powder, shot, and other supplies were stored for the ship's guns and cannons.
Gun Deck:  The gun deck is where the cannon were located.  On merchant ships, this deck was used to hold additional cargo.  In the Mayflower's case, the gun deck is where the passengers lived on the voyage to America.
Capstan and Windlass:  Large apparatus which were used to lift and lower heavy cargo between the decks.
Cargo Hold:  This is where the Pilgrims would have stored their cargo of food, tools, and supplies during the voyage.

The Gun Deck, sometimes referred to by the Pilgrims as "betwixt the decks" or the "tween deck," is where the Pilgrims lived for most of the voyage.  They occasionally ventured to the upper deck, especially during calmer weather when they would be less likely to get in the way of the seamen and there was less danger of being swept overboard.  The gun deck had about four gun ports on either side of the ship for cannon.  Even though the Mayflower was a merchant ship, it needed to be able to defend itself from pirates, and needed to be prepared for the possibility of conscription (when England was at war, the King or Queen could turn merchant ships into military vessels.)  The height of the gun deck was around five and a half feet.

During the voyage, the 102 Mayflower passengers lived primarily on the gun deck, or the 'tween deck.  The length of the deck from stem to stern was about 80 feet, of which about 12 feet at the back belonged to the gun room and was probably off-limits to the passengers.  The width at the widest part was about 24 feet.  Various hatches provided access to the cargo hold below.  The windlass and capstan, both used to haul heavy items by rope between the decks, also took up floor-space, as did the main mast in the middle, and the sprit sail mast in the front.  Many of the families built themselves small little "cabins," simple wooden dividers nailed together, to provide a small amount of privacy.  Others, especially the young single men, just took up any old spot--many found shelter within a shallop, a 30-foot sailing vessel that the Pilgrims brought with them, and which they had dismantled and stowed on the gun deck.  The two month voyage, with many young men living inside of it, caused considerable damage to the shallop, and cost the Pilgrims several weeks of time to fix after they arrived.