How may different Enterprises were there?
How many were featured in TV or film?
The last I heard there was a 'J' in the show 'Enterprise'.
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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 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.
Based on the episode TNG: Parallels, there were at least...
"Two hundred eighty-five thousand"
... ships called Enterprise seen (or referred to) within the main TV canon.
This video-clip offers a brief glimpse of around 300 of them.
Wikipedia mentions 8 + 4 from alternative timelines.
300+ Enterprises present in episode "Parallels"
Star Trek Online, which is canon with the Star Trek Universe, has NCC-1701-F, an Odyssey-Class Super-Heavy Cruiser. The Odyssey-class is essentially the Soverign class but with detatchable saucer section.
Images and information can be found here: U.S.S. Enterprise (NCC-1701-F).