All Saiyans and Half Saiyans can ascend all the levels of Super Saiyans. There are 5 levels, SSJ 1, SSJ 1 Ascended (which means you can remain in SSJ 1 for a long period of time), SSJ 2, SSJ 3, and SSJ 4. All the levels have a certain power level criteria like the rest of the transformations for the different races, the thing that is different is for Saiyans there needs to be some sort of trigger for them to become angry and ascend.
Triggered upon site of the moon, or an artificially made one, true Saiyans also have the ability to transform into a huge ape unleashing the only true power known to Saiyans before Goku's Ascension. Although uncontrollable at first, once in command the Oozaru form is a force to be reckoned with.
Humans have two different transformations that they can achieve, they are Hidden Power and Unlocking Potential. These two power ups increase the power of humans and allows them to reach new levels of power.
Namekians, like humans have two transformations that they can achieve, they are Hidden Power and Unlocking Potential. These two power ups increase the power Namekians posses and allows them to reach new levels of power.
Andriods have ways to increase their power levels as well, but instead of transformations, they are upgrades. There are three of them, Upgrade 1, Upgrade 2, and Upgrade 3. These three upgrades significantly increase the power of Andriods.
Cell has two transformations, one that occurs when he absorbs Andriod 17 and one that occurs when he absorbs when absorbs Andriod 18. Now these two things occur when Cell reaches the required PL and do not actually mean that Andriod 17 and 18 are necessart to be members or that they die. It is just the significance of Cell absorbing them that makes him tranform.
Cooler has two transformations, his normal form and his final form. Cooler begins with all his transformations and can go back and forth between them.
Freeza like his brother Cooler begins with all his Transformations. Unlike his brother he has 4 forms and can go back and forth between them with ease.
Zarbon has one transformation that he can perform. He goes from his initial state to a gruesome monster that increases his power level.
Majin is a transformation that people with Evil alignment can acheive. It requires going to Planet Zun to train with Babidi and having the correct power level. Demons are under the control of Babidi while in this form and are pure evil. Once strong enough they are able to break free of the control Babidi has over them.