(covers information from several alternate timelines)
The Olympic-class served the Federation as early as the 2370s. (DS9: "Sacrifice of Angels") In an alternate future, the Olympic-class was utilized primarily for medical emergencies. (TNG: "All Good Things...")
The Olympic-class featured a primary section-engineering section-warp nacelle layout common to most Starfleet vessels. However, unlike most Starfleet vessels, the Olympic-class featured a spherical primary hull, similar in outward appearance to the 22nd century Daedalus-class. Its deflector dish was incorporated into the lower forward quarter of the primary hull and was more of a strip than the more traditional shape. Its impulse engines were located on the upper third of the aft hull. A large shuttlebay was situated on the middle upper dorsal section of the secondary hull. (TNG: "All Good Things...")
In an alternate future timeline, the Olympic-class was equipped with sensors and defensive systems that were very limited, leaving the ship to be no match for the weapons of the Klingon attack cruisers of that era. In addition, it was capable of at least warp 13. (TNG: "All Good Things...")
- USS Pasteur (NCC-58925)
- USS Nobel (NCC-55012)
In an interview with Michael Okuda in Journey's End: The Saga of Star Trek: The Next Generation, it was stated, and shown on his computer-generated dedication plaque, that the USS Pasteur was of the Hope-class, named after the WWII hospital ship USS Hope (AH-7). It was later revealed, in the Star Trek Encyclopedia, that the name "Hope-class" had only existed in an early version of the plaque, and was later renamed as Olympic-class. This designation was taken from Bill George having originally named the Pasteur as the USS Olympic. 
An image of the Olympic-class, drawn by Doug Drexler for use in the Encyclopedia, which first appeared on an okudagram in "All Good Things...", also later appeared on a background bridge monitor in DS9: "Sacrifice of Angels".
Design and studio model
Roots of the Olympic-class' design were laid down when Bill George, over a multitude of years as he grew up, became fixated by one of the earliest design concepts that Matt Jefferies created for the original Enterprise, which George saw in the reference book The Making of Star Trek (pp. 81 & 82); the defining characteristic of this form of the original Star Trek vehicle was that it had a massive spherical primary hull, which Jefferies seriously considered at the time as he deemed the sphere the best possible pressure vessel for use in a vacuum, but eventually dispensed of as being "too bulky". (Star Trek: The Magazine Volume 1, Issue 10, p. 28)
While serving as a model maker at Industrial Light & Magic, Bill George suspected that a similar-looking ship might make a nice addition to the final season of Star Trek: The Next Generation and he contacted Michael Okuda to investigate this possibility. Okuda responded by expressing that TNG's production staffers required a starship that was contemporary to the Enterprise-D while still having a unique design. This led George to submit a sketch he did of a ship that was suitably contemporary and had a spherical primary hull. Though Okuda was initially noncommittal about whether the craft would be used on the show, he commented that doing so was a possibility.
Bill George thereafter worked on the weekends to build a studio model based on the artwork. He constructed the miniature around the same time as ILM began working on Star Trek Generations. (The Making of the Trek Films, UK ed., p. 167) "Bill made the model in his spare time as a labor of love," Visual Effects Producer Dan Curry noted. (Cinefantastique, Vol. 25, No.6/Vol. 26, No. 1, p. 64)
In the script for "All Good Things...", the Pasteur is described as "a small and sleek vessel with the 24th century equivalent of 'Red Cross' markings."  However, the model Bill George had built was used for the ship. "[He] called us up to ask if we would like to use it," stated Dan Curry. (Cinefantastique, Vol. 25, No.6/Vol. 26, No. 1, p. 64) George himself continued, "Producer Peter Lauritson ended up renting it from me." (The Making of the Trek Films, UK ed., p. 167) Normally, the series' policy forbade accepting unsolicited models. However, due to extreme time pressures at the time of production on "All Good Things...", Lauritson got dispensation in this one instance. (Star Trek: The Next Generation Companion, 3rd ed., p. 303)
The Olympic-class model has the distinction of being one of only two studio models that were neither commissioned by the studio nor designed or built by Star Trek's own production team (the other one being the Promellian battle cruiser}. Except for the name change from George's original configuration of the model as USS Olympic (NCC-58925) to its final form as USS Pasteur (NCC-58925), though its original name would be retained as class designation, very few further modifications were necessary before shooting the Olympic-class miniature at Image G. 
Several production staffers were pleased that Bill George's model was selected to represent the Olympic-class. "That was really thrilling to see that used," enthused George, "and I was really proud that it was the spherical primary hull that harkened back to that really early design." (The Making of the Trek Films, UK ed., p. 167) Dan Curry remarked, "Bill's model was perfect for the Pasteur, because the spherical front would hold more beds than a sleeker shape like the Enterprise. It's a very stately, peaceful looking ship which is exactly what we needed." (Cinefantastique, Vol. 25, No. 6/Vol. 26, No. 1, p. 64)