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 and refits which resulted in a different model being used to represent the same ship, there are 4 NCC-1701s and two 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 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. 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*. The NCC-1701-A would have been under construction while the original Enterprise was still in service, and thus was likely originally intended to have another name and number before the destruction of Enterprise; possibly the Mississippi or Missouri according to some sources. She was decommissioned shortly after the events in *VI*, largely due to battle damage (which was to be the fate of the NCC-1701 before Kirk stole her in *III*).
* The NCC-1701-B is an Excelsior-class ship 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, although it is assumed that the 1701-B was decommissioned or destroyed before the C was christened.
* 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.
* The NCC-1701-D, a Galaxy-class heavy cruiser, was commissioned nearly 20 years after the tragic loss of the Enterprise-C. She was featured in the entire *The Next Generation* series, and in *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 nearly any ship built by the major factions, 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, the Enterprise-D had been hauled out of mothballs by then-Admiral Riker, retrofit with an additional warp nacelle and upgraded weapons capabilities, and commissioned as a dedicated warship. I want to say that this reincarnation of D received a modified designation like "-DA" or "-DX" but I can find no proof. 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 likely under construction with another name planned before Viridian III). She is about half the general height of the Enterprise-D and thought to be 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. 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.