The term Ascended refers to all humans who were magically altered by a variety of magic but mostly celestial magic. After their Ascension, the Ascended not only retained all of their original magical abilities but also gained more of them. They can channel celestial magic through their connection to celestial entities but do not "gain" celestial magic themselves.
Targonian Ascended are mostly a melding of a Human and Targon Prime celestial know as an Aspect, which primarily retain their human appearance. Shuriman Ascended, known also as the god-warriors or sunborn, are humans magically altered by the Sun Disc of Shurima using the stolen knowledge of and the sun's power, their common form is that of anthropomorphised animals, however they are able to take form they wish during ascension. Baccai are Shuriman Ascended who failed to fully ascend, where as Darkin are Shuriman Ascended who drastically altered their bodies with hemomancy. Currently most darkin who are trapped in weapons an approximation of their original darkin forms.