A colony ship (or colonial transport) was any type of spaceship used for the transport of potential settlers -- as well as the equipment needed to establish a new colony (or supplement an existing one) -- to a given destination.
Vessels of this type varied in size and longevity. Arguably, the DY-100-class SS Botany Bay, launched from Earth in 1996, was the first Earth colony ship. (TOS: "Space Seed") Prior to the development of the warp drive, colonial transports were only capable of sublight velocities. As such, most were sleeper ships — outfitted with cryogenic or other biostasis capsules to lower the metabolic rate of the potential colonists so they could survive a decades- or even centuries-long voyage. An early expedition of this type was the vessel Forty Families. (TOS novel: World Without End)
After the development of warp drive, colonization efforts from Earth increased. Between the years 2063 and 2079, 33 extrasolar colonies in 20 star systems were established by warp-driven craft. (Star Trek novel: Federation) One of those colonies was Terra Nova in the Eta Casseopiae system, founded by settlers aboard SS Conestoga. (ENT: "Terra Nova")
Colony ships were not necessarily purpose-built as such. Many, especially in the first 200 years of Terran spacefaring, were converted from freighters or in-system passenger liners. (The Starfleet Museum: Bison-Class Transports)
An example of a purpose-built colony ship in the 23rd century was the Brisbane-class starship. The Brisbane and her sister ships were called "colonization cruisers" as their spaceframes had been developed from that of the Surya-class starships (which were alternately designated light cruisers or frigates). (Starship Recognition Files 2)
Another colony ship of the same era was the Cochrane-class colonial transport, which first entered service ca. 2254. (FASA)