This article contains spoilers for the most recent episode of Star Trek: Discovery. It will also have a lot of dates extrapolated from conjecture and from Star Trek ancillary publications. We Trekkies love this stuff.
After 15 episodes, a lot of angst, a good deal of death and violence, some genetic manipulation, and a visit to a parallel universe, Star Trek: Discovery has closed out its first season. And while the season's final episode wasn't the first part of a two-part cliffhanger (a long-running Star Trek tradition), it did end with a final and unexpected surprise. As the USS Discovery is heading to Vulcan to pick up its new captain, they are approached by another Starfleet vessel which has issued a distress signal. The makers of the show probably took a great deal of delight in tantalizing the audience with their dramatic reveal; there’s a brief, incomplete close-up of the incoming ship's call letters: NCC-17_ _.
And, lo, there it is, the USS Enterprise in all its glory.
To start: In 2256, the Enterprise had already been in commission for 11 years, having first launched in 2245. The first captain of the Enterprise, and the one who took it on its very first five-year mission, was a man named Robert April. To offer a fun real-world genesis: Robert April's name was derived from Gene Roddenberry's initial pitch for Star Trek back in 1964. Roddenberry suggested Robert April as the name of his show's potential protagonist. Through typical re-writes, April was eventually changed to Pike, then again to Kirk once the show was picked up. In a strange case of cosmic synergy, this production note leaked its way into Star Trek canon, and now Robert April is the official “first captain” of the Enterprise (and was even seen on The Animated Series).
What we knew previously: After five to seven years, Robert April handed the keys of the Enterprise over to Christopher Pike, who was in command from about 2251 until about 2263, having completed two full five-year missions (Then, according to what we know of the timeline, Pike was promoted, and Kirk took command of the Enterprise. The rest we've seen). I imagine retrofits and repairs account for the Enterprise's multiple-years-long bouts of downtime.
What we don't know: What Captain Pike was doing with the Enterprise from the time he took command (in the early 2250s) up to the events of the original Star Trek pilot “The Cage,” which took place in 2254.

Pike's crew included his first officer Number One (named Una in the Discovery novel Desperate Hours), chief medical officer Dr. Boyce, Yeoman J.M. Colt, navigator Tyler, and, of course, science officer Spock. Spock, who we now know is also the adopted brother of Discovery’s Michael Burnham, is the only character who made it from the original pilot to the first Star Trek series with William Shatner.
We now know that, sometime prior to the events of “The Cage,” the Enterprise had a run-in with a now-familiar Crossfield class vessel called Discovery. As a Trekkie with an obsession with continuity, I can assuage any worries you may have: There's no reason not to trust this. Everything we know about Captain Pike takes place during and after the events of “The Cage” (including the accident which robbed him of most of his physical faculties, as seen in the classic two-part episode “The Menagerie”), so it's perfectly natural to assume that the Enterprise could have fought in the same war as the Discovery, and it's most certainly reasonable to postulate that the two ships could have crossed paths.
Indeed, it's almost likely. The crew of the Discovery would certainly know of the Enterprise (in fact, Burnham mentioned the ship earlier this season). Unlike the Discovery, the Enterprise was not considered a technological marvel, and was not extraordinarily equipped, feeling a lot more like a submarine than a spacecraft. In the eye of the Discovery, the Enterprise was likely more seen as a reliable war horse. I imagine it would be like someone with a new Tesla meeting someone with a 1971 Dodge Challenger. One is a slicker machine, but the other is a classic.
There is one problem, of course: Since Star Trek: Discovery has been rather cavalier in its alterations of Star Trek's in-world technology, no matter how accurate we are with the timeline of the Enterprise, we sticklers for continuity (or “old whiners” as you may know us) will still have to make peace with the fact that this Enterprise is going to look very different than what we're used to, and it's going to have access to new tech that it didn't have in its original iteration.
But then, Star Trek: Discovery is also introducing us to an Enterprise that is younger than any rendition we've seen before. Perhaps the Enterprise has been upgraded since “The Cage” but will be wrecked and rebuilt at some point and it will, only then, resemble the ship we know from the Captain Kirk era. And all of our Constitution class ducks will be in a row. Until we learn for sure, though, it's merely a fun topic of academic nerd speculation.