Level 1
Captain's Cabin
This is Shepard's private quarters.
Points of interest: Personal Terminal, Fish Tank, Armor Locker
The Armor locker can be used to change Shepard's armor and appearance.
The Fish Tank (Aquarium) holds any fish that Shepard has purchased
The computer terminal on your desk lets you review your squad and change their appearance, as well as read in game email and watch some training videos.
A sound system lets you play some music. There is also a display case for miniature ships Shepard has collected.
Meetings with crew, as well as some romantic encounters, can also take place in this room.
Level 2
Combat Information Center (CIC)
Points of interest: Bridge, Galaxy Map, War Room, Vid Comm
This level has the Bridge, where you can talk to Joker.
A private terminal lets you view messages and training vids.
The Galaxy map lets you explore the various planetary systems. Ensign Traynor is also located here.
In the War Room, you can review your war assets, total military strength, effective military strength, and Readiness Rating, and see the War Map.
The Vid Comm can be used to communicate with the Alliance.
Level 3
Crew Deck
Points of interest: Liara's Cabin, Main Battery, AI Core, Medical Bay, Mess Hall, Starboard Observation, Lounge, Crew's Quarters
The Mess Hall is the main area, where the crew often gathers.
Visit Starboard Observation for a nice view of Space
In the Medical Bay you can talk to Dr. Chakwas or Dr. Michel. You can interact with the beds in the medbay to reassign power (reset powers) and reassign power bonuses. Note that the first respec is free, but additional respecs will cost you credits, so don't do power resets too often.
The Crew's quarters is where many of the Normandy's crew sleep. There is a model in this area - Destiny Ascension
The Lounge is where sometimes the crew hangs up and socializes.
Main Battery and AI Core don't have anything interesting, initially.
Liara has a big room in this level. Go to her cabin when you want to talk to her. There is also an Intel Terminal maintained by Dr. Tsoni's Glyph in this room which lets you access upgrades and reward options.
Level 4
Engineering
Points of interest: Engineering, Subdeck Stairwell, Starboard Cargo, Port Cargo
In the main Engineering Deck, you can speak to Engineer Adams.
There are also a couple of ship models to collect on the lower level of this deck, including a Tulian Cruiser and a Sovereign.
If you allowed the reporter Diana Allers, to board the Normandy, you will find her at the Starboard Cargo.
Level 5
Shuttle Bay
Points of interest: Armory
There is an Armor Locker here, used to change your armor and appearance, just like in the Captain's Cabin.
You can speak to Lieutenant Cortez, the Shuttle Pilot, here, as well as James Vega.
A Procurement Interface lets you access stores and buy items for a 10% markup. A Weapons upgrade interface lets you purchase weapon mods.
James is often working out. Look for a Normandy SR-1 model near him.
The weapon bench lets you change your weapon loadout.