They were in Starfleet service by the 2320s through the 2370s. (TNG: "Tapestry"; DS9: "In the Pale Moonlight") They were used on missions of deep-space exploration. (TNG Season 4, display graphic) The Vulcan National Merchant Fleet and the Maquis also utilized them as freighters and transports. These vessels were partially constructed out of dentarium alloy.
In 2364, a ship of the class, the USS Ajax, received Kosinski's experimental warp drive upgrades, resulting in a measurable increase in performance and engine efficiency. It was later suggested that the enhancement may have been attributed to Kosinski only straightening out simple engine inefficiencies of the old ship. (TNG: "Where No One Has Gone Before")
In 2368, fragments from the navigational deflector array of a Vulcan transport of this class were recovered in the Hanolin asteroid belt from a downed Ferengi cargo ship in crates marked "medical supplies". The debris was spread over one hundred square kilometers. After running a molecular pattern trace, the ship was later identified as the T'Pau.
That vessel, and two others were later used by the Romulans as a "peace" envoy sent to Vulcan. These vessels traveled from Romulus and across the Romulan Neutral Zone at warp 1; aboard the three vessels were over two thousand Romulan troops.
Following the Federation's discovery of the Romulans' true intentions, the invasion of Vulcan, the Romulan force retreated toward the Neutral Zone. Upon an attempted interception by the USS Enterprise-D, a Romulan Warbird destroyed the three Vulcan transports and all the troops on board. (TNG: "Unification I", "Unification II")
In 2372, Starfleet Command ordered the transport of several CFI replicators to Cardassia aboard a Vulcan freighter. It was later determined that the freighter was under Maquis control, and instead used the vessel to divert the replicators to the Demilitarized Zone and into their possession. Michael Eddington left aboard the Vulcan freighter as well, as he left behind his Starfleet career, and to join the Maquis full time. (DS9: "For the Cause")
The Apollo-class name has appeared on screen only in a Starship mission status chart in Star Trek: The Next Generation. In the Star Trek Encyclopedia (4th ed., vol. 2, p. 33). the Vulcan spacecraft T'Pau, a ship mentioned in "Unification I", was identified as a member of the Apollo-class. In an internal ship list, dated to September 26, 1990, the freighter Odin, with the registry of NGL-12535, was identified as a member of the Apollo-class. 
The Star Trek Encyclopedia (4th ed., vol. 2, p. 329), as well, identified one of the starships lost at Wolf 359 as the USS Gage, an Apollo-class starship, with the registry of NCC-11672. The ship was only mentioned in a scene cut from the aired episode DS9: "Emissary". In the scene the Gage was ordered to attack the Borg cube with the USS Melbourne and the USS Kyushu. 
The Star Trek Encyclopedia (4th ed., vol. 1, p. 36) stated this class of starship was "named for the sun god in Earth's Greek mythology, as well as for the spacecraft that first carried Humans to the moon."
The Apollo-class, according to Rick Sternbach who designed the T'Pau, was equipped with a conjectural type of warp drive called the annular warp drive by the Star Trek Encyclopedia (4th ed., vol. 1, p. 30). Instead of conventional nacelles, a ship equipped with this drive would have a large hoop-shaped field generator.
According to Issue #141 of the Star Trek: The Official Starships Collection, the Apollo class starship was approximately 470 meters in length and had a maximum speed of warp factor 9.6. The class could accommodate around 650 people. It was equipped with particle beam weapons and graviton-based deflector shields.
In apocryphal materials, there were several classes of Federation starships with the name Apollo-class:
- In The Price of Freedom sourcebook of the Last Unicorn role playing game, a class of light cruisers with a similar hull configuration to the Galaxy-class is identified as Apollo-class. They are 315 meters long and have sixteen decks. The class was in production between 2325 and 2358.
- In later Last Unicorn supplements, the Vulcan T'Pau is also identified as the Apollo-class, with five decks and a length of 148 meters.
- In the video game Star Trek: Legacy, a class of ships similar to the refit-Constitution-class, but with a nacelle above and below the primary hull and no secondary hull, is identified as Apollo-class (β) scout ships.