"What's happened to them? They look like Super Saiyans, but I don't feel the energy... It's like it's their natural state..."
Super Saiyan Full Power (超サイヤ人フルパワー; Furu Pawā Sūpā Saiya-jin), only ever referred to as Super Saiyan (超サイヤ人; Sūpā Saiya-jin), in-universe is the mastered state of the ordinary Super Saiyan form, which replaces it once it is attained.
The form has alternatively been referred to as Super Saiyan Fourth Grade (超サイヤ人第四段階; Sūpā Saiya-jin Dai-Shi Dankai) on occasion.
The form is still essentially the Super Saiyan form. However thanks to being able to use the form like a natural state, the user's eyes and facial features can become relaxed and even cheerful. Additionally, their aura alters somewhat, making the radiating energy become reduced in tandem, shown visibly by the aura becoming smoother into a steady flow of waves rather than rapid bursts of spikes. As it is still essentially the first grade of Super Saiyan, the general traits all remain the same, including this form multiplying the user's base physical and energy capacities 50 fold. The user's hair is often portrayed as yellowish white, rather than the iconic yellow-gold of a regular Super Saiyan.
Goku is the one to come up with the idea of mastering the Super Saiyan form into a natural state while in the time chamber, stating that they need to get rid of the instability and restless feeling. By conditioning themselves to remain transformed for extended periods of time, the Super Saiyan can desensitize themselves to the negative effects of the transformation. Their wild emotions are kept in check, allowing them to stay more focused, which results in a more strategic fighting style and pattern of thinking, and ki consumption is reduced to negligible amounts.
By mastering the transformation, the Super Saiyan is able to retain transformation almost subconsciously, allowing them to transform seemingly indefinitely. In the anime, they are even shown able to sleep in this state peacefully, which is shown to radiate a positive feel to the environment as even wildlife are attracted to them. In requiring less ki to contribute to transform or even maintaining this state itself, the maximum amount of ki can thus be focused into attacks and stamina use, with no adverse effects present in the second and third grades. Once this state was mastered Goku was able to keep up with Perfect Cell, something which the Super Saiyan Second Grades could not do. Vegeta notes the incredible side effects of the transformation by stating the feeling Goku or Gohan's Super Saiyan Full Powered form isn't the same feeling he gets around a normal Super Saiyan. A downside to this form is that with the Saiyan's power instinctively remaining at full strength, it requires much more focus and control over all their actions. While Goku and Gohan were drinking orange juice from glasses, they repeatedly broke the glasses from the difficulty of controlling their strength.
In Dragon Ball Super, once Goku and Vegeta mastered the power of Super Saiyan God, the god-enhanced Super Saiyan can evolve further to Super Saiyan Blue, but both can still transform into their regular Super Saiyan states, as in the Universal Tournament, but the Super Saiyan's appearance during this time looks altered somewhat, specifically Goku, in his regular Super Saiyan form, a single forward-facing bang of his hair appeared to jut out from the main bangs, somewhat resembling his Super Saiyan Blue hairstyle, presumably it is due to his absorption of the powers of Super Saiyan God, and achieving the Super Saiyan Blue form, (previously, when Goku is in his regular Super Saiyan form before he trained with Whis after he mastered the power of Super Saiyan God, and attained the Super Saiyan Blue form, he lacks this additional bang of hair, until then), but like the latter, this bang does not always appear, it is absent in some scenes. Both Goku and Vegeta's auras in this regular state were slightly altered, instead of smooth and flowing like it was in the Cell arc, but somehow a bit more jagged and spiked, becoming closer to the original version of the form, and the aura somewhat gains a slightly more goldish hue than how it was originally (like what Goku's aura looked like during his fight against Beerus).
Usage and power
While never actually stated, the other Saiyans appeared to have obtained this mastery later on. Goten and Trunks both easily attained the Super Saiyan form, and mastered it at a young age.
Their fusion form, Gotenks, seen late in Dragon Ball Z, easily gained the ability to transform into all three levels of Super Saiyan in the series within no more than a half-hour of training in the Hyperbolic Time Chamber (although he has only ever been seen assuming the first and third forms, since he bypassed the second form to ascend to the third level), with the combined energies of Goten and Trunks balancing out the high ki consumption of the final transformation.
During Goku's battle with Beerus as a Super Saiyan God in the movie and anime, Goku dropped out of the form, but somehow managed to retain the strength of his Super Saiyan God form in his regular Super Saiyan form. Goku used the god-enhanced form to battle Beerus. Later, Goku trained with Whis and after gaining enough power his god-enhanced Super Saiyan form evolved into the Super Saiyan God Super Saiyan form. As a result of his god-enhanced Super Saiyan form evolving into Super Saiyan Blue, Goku was able to go back to using non-god-enhanced Super Saiyan form at will, which he did so during the Universe 6 tournament.
During the Universe 6 saga, Vegeta and Goku's usage of this Super Saiyan form post-Whis training was referred to as "Chou Super Saiyan".
At the point after Broly - The Legendary Super Saiyan, the movies move into a time period when the Super Saiyan form has since been outclassed by the more powerful variants, the only users of the basic transformation seen from this point on are Goten, Trunks, their fusion of Gotenks, as well as Gogeta and Vegito.
During the events of Dragon Ball Z: Battle of Gods, Beerus and Whis watch a rendered version of the battle between Super Saiyan Goku and Frieza on Namek. Later, Goku uses the form during his battle against the God of Destruction Beerus, although he retains much of the power received from his Super Saiyan God form, thus being able to hold his own as a regular Super Saiyan. Gotenks uses the form when he challenges Beerus but is easily taken out.
Video game appearances
In all of its game appearances, Full Power Super Saiyan is simply called "Super Saiyan" and is treated as the same form.