(covers information from several alternate timelines)
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Spatial torpedoes were torpedo weapons carried aboard Starfleet ships during the 22nd century. Prior to the installation of phase cannons and photonic torpedoes on Earth starships, these torpedoes were the most powerful weapons in the standard ship's armament. (ENT: "Fight or Flight")
Spatial torpedoes were a somewhat ineffective weapon against an opponent with deflector shields, as the torpedoes would sometimes bounce off the shields at impact and detonate. Also, the torpedo engine made the weapon so slow that in some cases it could be shot down with point-defense particle weapons prior to impact. (ENT: "Fight or Flight") The range of the torpedoes was fifty times shorter than with the photonic torpedoes. (ENT: "The Expanse")
The warhead of a spatial torpedo could be removed and utilized as an explosive, though moving an armed warhead around was not recommended. (ENT: "Future Tense") The strength of the explosive yield could not be altered. (ENT: "The Expanse") One type of spatial torpedo was the Triton class. These torpedoes were similar to the Romulan mines that Enterprise NX-01 encountered in 2152, deployed in orbit of a planet in a star system annexed by the Romulan Star Empire. (ENT: "Minefield")
The loading and firing of spatial torpedoes was partly a manual process. Inactive torpedoes were stored on racks in the armory. When a ship engaged in combat, the torpedo was removed from its holding rack and placed on a launching rail, which inserted the torpedo into its firing tube. The torpedo could then be fired, either from the tactical console on the bridge or the firing console in the armory. (ENT: "Fight or Flight")
Spatial torpedoes were part of the original armament inventory of Enterprise, and could be launched from six points on the ship's hull: four forward tubes, attached to the armory, and two aft. Even though the ship was equipped with photonic torpedoes prior to the starship's mission to the Delphic Expanse, the Enterprise kept its complement of spatial torpedoes. (ENT: "The Expanse")
The name and type of the torpedoes was never stated on screen. They were referred to as conventional torpedoes in "The Expanse". The association with the spatial torpedoes mentioned in "Minefield", is based on Startrek.com using that term of these torpedoes as well.  According to Startrek.com, the spatial torpedoes also contained more contact-explosive yields and ingredients than the photonic torpedoes. 
The Romulan mines noted to be similar to Triton class spatial torpedoes in "Minefield" were armed with tricobalt explosives. According to TOS: "Balance of Terror", Starfleet used atomic weapons in the Earth-Romulan War that began in 2156. This suggests spatial torpedoes might fall into this category, as they were still seen stored in the armory as late as season 4 of Star Trek: Enterprise.
The spatial torpedoes were designed by John Eaves. He sketched out various possible configurations, ranging from a conventional ICBM to the highly futuristic. Executive Producers Rick Berman and Brannon Braga ultimately opted for a design that was somewhere between these extremes. (Star Trek: The Magazine Volume 2, Issue 7, p. 55)