In Japan, the original manga is, and always has been, Dragon Ball. (The original Japanese title is the phonetic English words "Dragon Ball,.") However, when the series was optioned for TV, things got complicated.
The original Japanese anime series, based on the manga, was also called Dragon Ball. It ran from 1986 to 1989, and covered a little past the ending of manga volume 16, up to the end of Goku's final battle with his first archenemy, Piccolo. At this point, Toriyama was gearing up for a major new story arc, and Toei Animation decided to relaunch the anime under a new name. They chose the name Dragon Ball Z (pronounced "Zet" in Japanese). The reasons for the name change are obscure, but Toriyama joked in an interview with Banzai!, a now-discontinued German manga magazine, that he chose the subtitle "Z" because he was getting tired of drawing Dragon Ball and the last letter of the alphabet would make readers think the end of the series was approaching. (It's an open secret that Toriyama intended to end the Dragon Ball manga years before the actual ending, but was pressured into continuing it since it was such a moneymaker.)
No such luck; the Dragon Ball Z TV series ran from 1989 to 1996. At this point the manga series had already ended, but the licensors decided to continue the series in an anime-only form based on some of Toriyama's ideas and character designs. The resulting new series, in which Goku travels through outer space and meets a lot of aliens, was titled Dragon Ball GT ("Grand Tour"). Dragon Ball GT ran from 1996 to 1997 and is not considered canon by many fans of Dragon Ball and Dragon Ball Z.
Imagination-rich and information-poor fans spread rumors of other spin-offs with names like Dragon Ball AF. None of these are real, except for the 2009 Dragon Ball Kai, which is not a new series but a remastered version of Dragon Ball Z. (See "Isn't Dragon Ball just a bunch of speedlines and ripped dudes with bad hair screaming "It's over 9,000!"".)