There are several, including a number of non-canon ones from movies/filler/GT/etc.
Great Ape: Any Saiyan with a tail who sees moonlight or the light from a Power Ball is capable of this. Examples we see in the series and movies include Goku, Gohan, Vegeta, and Bardock and his team. Numerous other Great Apes are shown in anime-only flashbacks. Raditz and Nappa do not use the transformation in the anime or manga, but can do it in Budokai Tenkaichi 3 where a moon is available.
Super Saiyan: The form that Z really heavily popularized. It takes a pure heart, a high enough power level, and often some sort of traumatic event or distress to be pushed to reach this form. Goku, Trunks (both future and preset), Vegeta, Gohan, and Goten all achieve this form, as do their respective fused forms (Gotenks, Gogeta, Vegito.) There are several variations on the "first" Super Saiyan form, including the "second grade" form with slightly bulked up muscles (Vegeta calls himself "Super Vegeta" in this form), and the "third grade" form (often dubbed things like "Ascended Saiyan, Ultimate Super Saiyan, or Ultra Super Saiyan), which further bulks the character up to higher levels of attack and defense, but at the expense of speed as well as rapid power drainage, making it an impractical form to use in battle. Goku, Vegeta, and Future Trunks were all capable of the "third grade" form, but only Trunks used it, Vegeta and Goku knowing about its limitations.
Super Saiyan 2: The next main Super Saiyan form. This is similar to the "ascended" versions of Super Saiyan, but with all abilities balanced and no negative limitations. Gohan was first to achieve this form, but Goku and Vegeta later did as well, presumably in the 7 years between the Cell and Buu Sagas. It is implied that their fusions would be capable of this form as well, as Gotenks can apparently be Super Saiyan and Super Saiyan 3 and "skip over" the form.
Super Saiyan 3: The only non-fused character capable of this transformation is Goku. It is far more powerful than the previous states, but rapidly drains the user's energy. Gotenks can achieve the form as well. Several "what if" scenarios occur in video games, showing Vegeta, Future Trunks, and Broly in this form.
Controlled Super Saiyan/Legendary Super Saiyan: The forms used by Broly, the Legendary Super Saiyan. No other character is shown using them.
Golden Great Ape: A transformed, golden "Super Saiyan" form of Great Ape. Achieving this form is necessary to reach Super Saiyan 4. In old Saiyan legends Vegeta talks about, he mentioned how the last Super Saiyan could only be one in the transformed state, meaning he is likely talking about Golden Great Ape. Goku and Vegeta are shown in this form.
Super Saiyan 4: A new form of Super Saiyan seen only in Dragon Ball GT. Appearance is radically different from previous forms of Super Saiyan, and the user must first achieve the Golden Great Ape form. This form, apparently, takes the user to their body's "prime", meaning even Goku turned into a child achieves an "adult" form. Only Goku and Vegeta are seen in this form, as is their fusion, Gogeta.
Super Saiyan God: A new form introduced in the Battle of Gods movie, which only Goku reaches. As the movie has not been dubbed at this time, little is known about it.