False Super Saiyan (Giji Sūpā Saiya-jin) is a transformation that Goku takes during his battle against Lord Slug in Dragon Ball Z: Lord Slug.
In this form, Goku's irises and pupils are no longer visible, mostly in turn to the power and bleak rage. His body bulks up a little, his hair straightens like that of a regular Super Saiyan, but there is no considerable change in feature other than a red tint to his hair. He gains a yellowish surge of aura, somewhat but not quite resembling that of a normal Super Saiyan's. In turn, the form somewhat resembles a glorified Kaio-ken form, but there are notable differences.
When he transformed into the true Super Saiyan transformation during the fight against Frieza, Goku witnessed the murder of Krillin, something that drove him over the edge. This ultimate pain was not fully present during the fight against Slug, which is most likely why Goku did not transform fully.
During the battle against Lord Slug, Goku's strength began to quickly debilitate. Having no more options, Goku's fury surged, causing him to gain a massive power increase, enough to quickly overwhelm Slug. However, after a quick sequence of Goku thrashing Slug around, his power quickly diminished. It is safe to say that in a nutshell, the False Super Saiyan transformation occurs when a Saiyan has the required physical power but lacks the mental need to transform into a Super Saiyan. It is apparent however that the form gives a large power increase, enough for Goku, who was being easily beaten by Slug, to quickly overwhelm him. The power increase is never stated nor looked into, but it would be suggested that it at the very least, multiplied the user's power by around 10, seeing as how even Goku's strength enhancing Kaio-ken techniques were futile against Slug.