As a succinct answer, there are 8 warp-capable starships with the name Enterprise that have been seen in at least one episode of a Star Trek series or movie to date. If you count alternate timelines/universes, and refits which resulted in a different model being used to represent the same ship, there are 4 NCC-1701s and 2 NCC-1701-Ds, bringing the total to 12.
Prior to the events of the Star Trek canon, in real history, six tall (sailing) ships, two aircraft carriers, and a space shuttle have borne the name “Enterprise”. They, plus a fictional sublight interstellar ship from the 2100s numbered ECV-330, are paid homage to in Star Trek: The Motion Picture and in other ways through the canon.
- The NX-01, best described as a light cruiser (slightly smaller than a Miranda class), was the first faster-than-light capital-class starship to be built by the human race. It is the ship for the series Enterprise, and established the Humans as a space-faring race, paving the way for the United Federation of Planets to be formed.
- The NCC-1701, a Constitution-class heavy cruiser, was the ship for the entire original series and, in refit form, for the first three movies. NCC-1701 was destroyed by Kirk in Star Trek III to keep it, and information about the Genesis Experiment, from falling into Klingon hands.
- The TOS episode “Mirror, Mirror” shows a nearly identical NCC-1701 to the one from the rest of the serires, from an alternate universe in which the Federation is instead the Terran Empire. This ship is named the ISS Enterprise instead of the USS Enterprise, and is captained by a “mirror” James T. Kirk.
- NCC-1701 was extensively overhauled from the version seen in the original series after the end of TOS and before the events of Star Trek: The Motion Picture, and was also seen in that form in Star Trek II and Star Trek III, but (in-canon) the movie version was considered the same ship as the one in TOS.
- An alternate version of NCC-1701 is seen in the 2009 Star Trek movie; the entire plotline of Star Trek follows an alternate timeline to the events that would have preceded TOS.
- The NCC-1701-A, a refit Constitution-class ship, is unveiled at the end of Star Trek IV (the majority of IV having taken place with Kirk captaining the hijacked Klingon Bird of Prey from III), and is used for V and VI. According to most canon sources, the Enterprise-A was originally the Yorktown, NCC-1717, which was quickly refit after being disabled by the Whale Probe in ST IV, and recommissioned as NCC-1701-A as a gift to Kirk and his crew for their service to the Federation. The Enterprise-A is assumed to have been decommissioned shortly after the events in VI, largely due to damage sustained in the battle with General Chang over Khitomer (which was to be the fate of the NCC-1701 before Kirk stole her in III).
- The NCC-1701-B is an Excelsior-class battleship featured in the introduction to Star Trek: Generations, which occurs very late in James Kirk’s life. He is killed on the maiden voyage when the ship is called on to rescue two shiploads of refugees who have been caught in an energy ribbon. The fate of the Enterprise-B herself is unknown based solely on series and movies, but licensed novels indicate the ship was lost and presumed destroyed after its final transmission following the Cardassian invasion of Bajor, indicating the crew had contracted some unknown infection.
- The NCC-1701-C is an Ambassador-class heavy cruiser, seen in one TNG episode, “Yesterday’s Enterprise”, where the ship passes through a temporal rift into the time of the Enterprise-D. The Enterprise-C was destroyed with the presumed loss of all hands in a battle against four Romulan warbirds while defending a Klingon outpost, a mere 5 years after her reported christening date in 2339.
- The NCC-1701-D, a Galaxy-class heavy cruiser, was commissioned nearly 20 years after the tragic loss of the Enterprise-C. She was she ship for the entire The Next Generation series, and for Star Trek: Generations. The Enterprise-D was one of the few Enterprises to be large enough to house spouses and children of the crew aboard, which Captain Picard apparently tolerates. As such, it was the largest of the Enterprises in the main canon (what is known about the Enterprise-J suggests it would dwarf any ship built by the major factions during the main canon, including the Enterprise-D). She was destroyed during the events of ST:G while in orbit over Veridian III in a battle with rogue Klingon agents.
- In an alternate future timeline seen in the TNG series finale “All Good Things”, the Enterprise-D had been hauled out of mothballs by then-Admiral Riker, retrofit with an additional warp nacelle and upgraded weapons capabilities, and recommissioned as a dedicated warship. This alternate future in fact never came to pass in the “main” timeline.
- The NCC-1701-E was a Sovereign-class cruiser launched in 2372, one year after the Enterprise-D’s destruction (and thus, according to First Contact‘s writer, her keel had been laid while the Enterprise-D was still in service, and so was originally planned to have another name). She is indicative of the latest wave of ship classes after the era of the Galaxy (such as the Intrepid and Nova classes), with an egg-profile saucer and minimal “neck” joining the saucer to the aft section. As a result, she is about half the height of the Enterprise-D and slightly smaller overall, with no provisions for families or children of crew members aboard. She is featured in all TNG-era movies after Generations (First Contact, Insurrection, Nemesis). She was critically damaged in Nemesis, but after extensive repairs in drydock she remains active as of the end of the known timeline.
- As an aside, while the ship named Enterprise has kept the designation NCC-1701 through the eras, the majority of other Federation ships have been numbered in normal sequential order, with the result that the U.S.S. Voyager, which is roughly contemporary to the Enterprise-E, has the designation NCC-74656. One can assume from her numbering that roughly 73,000 Starfleet capital ships have been commissioned between TOS and the Voyager arc, of which maybe half are still in service. Space is a dangerous place.
- Versions F-I are never seen or discussed in canon but are assumed to exist due to the existence of the NCC-1701-J. The NCC-1701-J appears in an episode of Star Trek: Enterprise as a ship from the 26th century of a possible future timeline. Its class, and much of its specifics, are unknown, but it was conceived to be a truly massive ship (so much so that turbolifts would have been replaced with transporters) with capabilities far exceeding ships in the known Star Trek canon universe.
The NX-01 was featured in Enterprise.
NCC-1701 was featured in TOS (and the first 3 movies)
NCC-1701-A was featured in TOS-era movies after 3.
NCC-1701-B was featured in Star Trek: Generations.
NCC-1701-C & D were featured in TNG (C for one episode, possibly a 2-parter)
NCC-1701-E was featured in TNG-era movies
An episode of Enterprise featured a small scene on the NCC-1701-J, from which we can assume the existence of F, G, H, and I.
Alternate versions of each ship likely exist for each ship, though only Alt-NX-01 and Alt-NCC-1701 were seen on screen, I believe. Alt-Alt-NCC-1701 was featured in the Star Trek franchise reboot movie.
This list is from memory, and not exhaustive, and does not cover questionably-canon sources like RPG manuals, comics, or novels.