(covers information from an alternate timeline)
The Enterprise-J's exterior was seen only in the background in the form of graphics on a computer screen. "We originally had an exterior shot showing the Enterprise-J in its glory in the middle of the battle," recalled Manny Coto, "and we had to drop it because of money." Both the visual effects department and Coto himself regretted this outcome. (Star Trek: Communicator issue 152, p. 50) Following the omission of the exterior shot, the look of the Enterprise-J, aside from the ship's "observation deck", was essentially not at all described in the final draft script of "Azati Prime", nor was even the computer display of the craft. However, the same script did specify that the Enterprise-J was to be armed with "a futuristic particle beam weapon".
A CGI model of the ship was built by production illustrator Doug Drexler. According to Robert Bonchune, "From what I understand, it was done quick and dirty, so was not really detailed in any way like a model we would use in multiple shots."  Drexler himself explained, "With two days before a production meeting [I had] to think fast and not obsess… especially since I wanted [Production Designer] Herman to be able to show a rendered animation of the ship in flight [....] If we had a few weeks, it would have finalized differently. As it was, the model was bashed out in a matter of hours." (X)
One concept for this type of ship that came under consideration was the Altair-class, which had previously been submitted (without being approved) for both the USS Voyager and Enterprise NX-01. Doug Drexler recalled, "I would sleek it somewhat, and rebuilt it in Lightwave. No luck that time either (although the engines would end up on the approved J ship)." (X) A computer-animated sequence showing the exterior of this design while in spaceflight was generated while the idea of using it as the Enterprise-J was still being considered. (The sequence can be viewed here(X)). (X) Once the approved Enterprise-J saucer section was mated with the Altair-class wingship, the design became known as the Congo-class, which turned out to be another unapproved runner-up for the eventual Enterprise-J design. (X) "I opted for spindly nacelle struts because I felt it suggested a technology beyond what we were familiar with," Drexler explained. (X) While designing the craft, he didn't have enough time to determine such aspects as the number of decks and the crew complement on board the ship. 
Set Designer Lee Cole was involved in planning the Enterprise-J's interior, a task she considered "one interesting thing I did." She went on to relate, "That was really startling when they said 'OK, you're going to do a future ship that goes even further into the future.' We're already in the future, and now you have to think of what the future looks like 400 years ahead of Archer's future!" (Star Trek: Communicator issue 152, pp. 42-43)
Ultimately, the appearance of the Enterprise-J pleased Manny Coto. "I thought the design was very sleek and cool, too," he enthused. (Star Trek: Communicator issue 152, p. 50)
Although logic would suggest this was the eleventh Federation Starfleet vessel to carry the name, Doug Drexler speculated that was not necessarily the case, commenting, "In six hundred years, they may have expanded on the alphabet and 'J' may mean something that is not the same as what we think it is, so we have to take everything with a grain of salt here." 
An official class name for this type of ship has not been confirmed, although Doug Drexler approves of "Universe-class". (X) "That's what I've always called it," he stated. Drexler believed "Universe-class" made sense, considering that the USS Enterprise-D was a Galaxy-class ship. He more specifically chose the name because "it suggests that its mission envelope encompasses the universe – you know, it's not just our galaxy – [and] [...] that they might be able to fold time."  That is indeed part of how Drexler has imagined the vessel. "They can fold space, and they are exploring other galaxies besides the Milky Way," he hypothesized. (X)
In the Star Trek Encyclopedia (4th ed., vol. 1, p. 244), the Enterprise-J is identified as a Universe-class starship. The same reference book additionally described the ship as having an overall length of 3219 meters.
Doug Drexler had estimated the length of this vessel as being about two miles. (X) "I don't think it's that large [in the scheme of things], really. I run into this a lot on the internet, where people freak out. They say, 'Two miles?! That's absurd!' But if you are a reader of science fiction and you don't just watch science fiction on television," Drexler said, with a laugh, "you'll know that huge, enormous ships is not a strange thing; I mean, there are ships that are many times bigger than that. And even in the Star Trek universe, we've seen alien races that can build machines like a Dyson sphere, for instance. Two miles seems like a lot in Star Trek terms, but the whole idea is that the J is supposed to be, you know, [from] a far-flung future date, so that means you've gotta stretch things to the point where people are going, 'That's ridiculous!' because that's what makes it futuristic, is it's hard to believe." 
Doug Drexler theorized that the hull of the Enterprise-J could have been organically grown around a skeletal frame. 
Doug Drexler has deemed the Enterprise-J to be a "multi-generational" ship. (X)  "Off the top of my head, you know, a population of a million [including families] doesn't seem out of line, really," he commented. 
Doug Drexler thought of the Enterprise-J as having "large parks, entertainment zones, and entire universities on board. The ship is so large that turbolifts would be replaced with site to site transporters." (X) Drexler has approved the idea that turbolifts were used as a backup on the craft, in cases when the site-to-site transporters were not functioning. "You might even have freeways on the ship, where you can drive a car from one end to another," he said. 
Doug Drexler was extremely doubtful that the layout of the craft would include a traditional-looking bridge. He thought the ship's computations wouldn't be done via bio-mimetic gel packs or anything similar to those. "For all we know, the ship is using, you know, the synapses of every person aboard the ship to do its computations," he speculated. Drexler thought the vessel wouldn't include any holodecks, since they would be deemed superfluous, as all recreation could instead take place via neural interfaces which the members of the crew had inside their heads. He hypothesized, too, that the vessel could be operated with the same method, by crew members who might stay in their cabins while remotely operating the craft with the implants in their brains and with essentially floating tactile interfaces which surrounded the user and were built-in, similar to computer systems portrayed in such films as Minority Report and the Iron Man film series. He thought the ship might consequently be self-aware and jokingly suggested that, instead of featuring a bridge aboard the ship, the craft might contain just a room with a toilet. However, he conceded that wouldn't be very interesting for audiences to watch. "If there are consoles and there are [more traditional] interfaces, we're doing it basically because it's a legacy thing," he reckoned. "You may be doing all the interfacing with the ship in your head, but it would probably be more fun to sit at a console." He also speculated that the interior of the ship, which he didn't approve of, was actually built for Archer, just so he could relate to it, and that the craft might be capable of communicating through time. "But the number one purpose of the ship is... exploration, and exploration of the universe and ourselves; the more we learn about the universe, the more we know about ourselves, and that's the mission of the ship," Drexler mused. 
When interviewed years after ENT finished, Doug Drexler was welcoming of fan theories regarding the Enterprise-J but also conceded, "I think probably anything anyone has thought of is not advanced enough." Though Drexler has heard some Star Trek fans complain that the struts were too spindly to support the ship's nacelles, Drexler rejected that criticism. As for the other theories about the vessel, he concluded, "I would love to see, you know, these really far out ideas for the Enterprise-J be what it's all about, but I think that if we did it, it probably would have been pulled back a little more. You know?" 
The Universe-class was introduced in Star Trek Online expansion Agents of Yesterday, with the appearance of the Enterprise-J. It was later made playable as the "Universe Temporal Heavy Dreadnought Cruiser".