Ships of the class
The Springfield class was a design study by Ed Miarecki and Michael Okuda, and the filming model was built as a kitbash by Miarecki for the "graveyard" scene in "The Best of Both Worlds, Part II".  The design shares a resemblance to the Cheyenne-class design, and was constructed using parts from the smaller scale AMT/Ertl USS Enterprise-D model kit saucer section, a bridge module from the larger Enterprise kit, and commercially-available highlighter markers for warp nacelles. Okuda would later remark on the model:
"We did the U.S.S. Chekov model because Riker (or was it Shelby?) had a line in the script in which he mentioned the ship in the graveyard. The line was filmed, but later, someone thought the reference to Pavel was too cute, so it was changed in postproduction to Tolstoy.(...)The Chekov was built by Ed, based on the small Ent-D model saucer, with two nacelles and marker pen engines. Unfortunately, I can't release these photos, but I think they'll eventually be in print.(...)[The nacelles were] Not glued directly onto the saucer. I think Ed made some pylons from sheet styrene.(...)The Melbourne, Kyushu, and Chekov were specifically labeled as such because of the script references to those ships. I didn't know about the re-voiced name (Tolstoy) until I saw the final episode on the air, so I didn't do a label for that ship."
The model of the Chekov was labelled "NCC-57302", as was shown on the production photos Okuda referred to (which were used at a slide-show presentation at a convention in 1991) though the Star Trek Encyclopedia (3rd ed., p. 471) gave "NCC-53702" as registry, which might well be a typographical error.
In the same publication Okuda also made an attempt to remedy the Chekov/Tolstoy-discrepancy by assigning the USS Tolstoy as belonging to the Rigel-class (β). At the time of filming of the episode the class designation was not in use as he remembered, "Rigel: I don't think we ever came up with a ship design specifically tied to that name." It was unclear if the renaming at the time meant that the filming model was intended to be the Tolstoy or that a reference was made to a ship not seen in the episode, though Okuda apparently proceeded eventually from the former assumption. He himself reinforced that notion when he gave a screen-used piece of wreckage he owned, in consignment to the Propworx' STAR TREK auction of 4 June 2011, described as, "Irregular piece of the saucer section of the U.S.S. Chekov from the "starship graveyard" scene in the Star Trek: The Next Generation episode "The Best of Both Worlds, Part. II."" (Propworx Star Trek auction catalog, page 44) The piece, Lot 42, estimated at $100-$200 sold for $400.
According to Issue #110 of the Star Trek: The Official Starships Collection, the Springfield-class was a type of frigate with a crew of 430, a length of 325 meters, and a top speed of warp 9.2. This class was used for deep space exploration and defensive patrol duties. This class was also equipped with eight phaser arrays and two photon torpedo launchers. Around the time of the Dominion War in 2375, the remaining Springfield-class ships were slated to be decommissioned and replaced by more combat-capable starship types.