There are many different types of apartments in New York City and a lot of the terminology can be quite confusing to the newcomer. Apartments are initially categorized by the age of the building, either pre-war (World War II, that is) or post-war.

Pre-war Apartments:
Aficionados of the pre-war apartment go for the high ceilings, hardwood floors, crown moldings, arched doorways and other old world charms that one usually finds in this type of apartment. Those requesting a pre-war often refer to post-war apartments as "cookie-cutter" and lacking in "charm" and therefore insist on a pre-war apartment. Pre-war buildings tend to be of sturdy construction offering thick firewalls between apartments and therefore can be quieter than the dry walled apartments found in more modern buildings. Watch out for the creaky floors, though. You can sometimes hear every step of the neighbor above in a pre-war apartment.

Post-war Apartments:
For those who like new and modern, the pre-war just doesn't cut it. Many post-war buildings, especially those recent construction, offer amenities and finishings often sought after by many young tenants. What post-war buildings lack in "charm" can be made up for in perks: a health club, swimming pool, meeting/party room, modern appliances, etc.

The next way to break down New York City apartments is by service level, as in whether it is a doorman or a non-doorman building.

Doorman Buildings:
Usually, if there is building employee stationed at the door or in the lobby(doorman, concierge, etc), the building is referred to as a Doorman Building. Technically, there should be a uniformed attendant to open the door for residents and guests for the building to be designated a "doorman building" but these days the term is used loosely. If there is no one assisting you with the door but there is a building employee in the lobby at a desk (concierge) then the building is traditionally referred to as an "attended lobby building". If there is both a doorman and a concierge, the building is often designated a "full service building". Doorman buildings can also offer a part time doorman only.

Non-doorman Buildings:
Non-doorman buildings come in two varieties: elevator building and walk-up.

Brownstone:
Early 20th century single-family homes, usually four or five stories high. The name was derived from the brown stone materials used in construction. The classic brownstone has an entrance several steps up from the street and a parlor floor with high ceilings and tall windows. The garden floor is a few steps down from the street and opens to the backyard. Today, many of the single-family brownstones have been converted into multiple apartments with separate entrances.

Town House:
An upscale single-family private home, usually three or four stories high, with a private entrance directly from the street. Many of these structures contain very desirable amenities including private backyards and roof decks, working fireplaces, and ornate hardwood floors. While some town houses remain as single-family dwellings, many have been reconfigured to accommodate multiple apartments.

OTHER COMMON TERMS

Live/work:
Usually refers to a commercial building where some residential tenancy is accepted. A live/work space has a kitchen and full bathroom with shower.
Commercial Loft:
Zoned for business use only. This loft usually has no kitchen and a small bathroom with no shower.
Sleep Loft:
This is an add-on structure usually found in apartments with high ceilings (10 to 14 feet or more). Typically it's like the upper of a bunk bed, with a ladder leading up to it. It usually accommodates at least a double-size mattress, although some are larger. The space below is often used as a closet or desk space, and the higher the ceiling, the more comfortable a sleep loft will be to sleep in.
Furnished Apartments (for rentals):
Can include light furnishings (basic furniture) or "soup to nuts" (dishes, utensils, small appliances, etc.)
Balcony or Terrace:
Often used interchangeably to indicate an outdoor railed platform extending from the apartment. It can be small enough to accommodate one or two persons standing or large enough for a picnic table, lounge chairs, and plants. It usually has a roof provided from the balcony/terrace of the apartment above.
Roof Terrace or Roof Deck:
This implies a larger terrace or patio provided by the roof of the extended floor below. It usually has brick wall barriers around the perimeter, is open to the sun, and some wrap around the corners of the building.
Garden or Backyard:
Many ground floor apartments offer private access to the backyard. It can be tiny or large, and in larger buildings with multiple rear exits, it may be semiprivate or shared.