Commisioned in 2293, the USS Enterprise NCC-1701-B was the third Federation Starship to bear the name Enterprise.
Design
We've known since Next Gen premiered in 1987 that the Enterprise-B was an Excelsior Class ship (from the models in the conference lounge). I would imagine that, following the badass appearance of the USS Excelsior in Star Trek III and IV, it seemed like a logical successor to the awesome Enterprises that came before. The Great Experiment, transwarp, you know, top of the line shit. However, by the time the ship made its on-screen debut in Generations, it was apparent that you couldn't really get more run of the mill than an Excelsior Class. These things are still in service after 94 years? Fighting in the Dominion War? Can you imagine the United States invading Iraq with a bunch of Wright Flyers?
So the Enterprise-B was revealed as a slightly new model of Excelsiors. It had extra impulse engines, redesigned warp nacelles, an expanded stardrive section, and a fresh paint job. The changes were welcome, but still leave an overall disappointing impression.
Maiden Voyage
Though the ship was launched with much fanfare, Kirk enthusiasts across the Federation were all a-titter at the appointment of the whiney, inexperienced Captain John Harriman to the Flagship. This prompted Starfleet to send half the bridge crew from the previous Enterprise along for the ride on the ship's shakedown jaunt across the Sol System.
Flying straight into the plot, the Enterprise soon receives a distress signal from a pair of transport ships carting a handful of El-Aurian refugees from their Borg ravaged homeworld. Harriman immediately tries to chicken out, but it turns out his is the only starship in range (they're still within Earth's solar system, so explain to me how that makes any goddamn sense). So off they go, and find the transport ships trapped in yonder energy ribbon. There is (naturally) some sort of interference preventing the transporters and tractor beam from working, so in goes the Enterprise.
Scotty is successful in getting a handful of survivors off the transports, but the Enterprise is now trapped within the rift. Harriman, in a stunning display of balls, volunteers to perform the necessary modifications to the deflector dish and get them out, but Kirk insists that the Captain stay on the bridge and heads down to the badly damaged engineering decks himself. He is of course successful (this is Kirk we're talking about), but on the way out, the Enterprise is struck by an energy discharge, rupturing the hull and sucking him into the Nexus. He is presumed dead, but we know better than that.
The Later Years
Now, nothing more is seen of the Enterprise-B in canon sources, but the ship is mentioned a few more times in Star Trek fiction. Captain Harriman stepped down in 2311 due to his involvement in the conspiracy surrounding the Tomed Incident, nearly sparking a war between the Federation and the Romulans. A promoted Demora Sulu, daughter of everyone's favorite original Enterprise helmsman Hikaru Sulu, took over command.
Sulu took a year long leave of absense in 2315 to care for her terminally ill grandmother. In her absense, the Enterprise embarked on a wildly successful mission of exploration, earning the (unknown) captain a promotion. Sulu then returned to see the ship into its final years.
The Enterprise-B's ultimate fate is unknown in both canon and non-canon works. It was presumably lost or decomissioned sometime before 2332, the year the USS Enterprise-C was launched.