Powers refer to many of the supernatural abilities the people of Olathe inherit. It's typically unknown how powers are acquired, or why they're there, as the game gives little to no explanation to it. The only information we can glean is inferred or, possibly, unofficial.
In LISA: The Painful, Brad Armstrong, Rando Armstrong, and Mad Dog have the ability to throw fireballs. Nern Guan and Ajeet Mandeep can shoot lasers from their fingers. Additionally, Rooster Coleman has some sort of Kinesis blast (Though that might be unrelated), Buffalo Van Dyke can summon some sort of spirit buffalo, and Jack can literally pull things out of his hat (Might also just be trick magic). The only information for why these people have these abilities can be gleaned from their description and an observation of their character.
The description for fireball attacks tends to read "a blast of flaming emotion" (Or something similar). Most characters that use these attacks have serious emotional damage or trauma, such as Brad's incredibly troubled life or the loss of Mad Dog's children. Additionally, both Brad and Rando are masters of the Armstrong style of karate, so it's possible that weaponizing emotions like this is something that could be taught, though it could just be a coincidence.
Very little is known about Finger Beams. The only users of finger beams in Painful are Nern and Ajeep, both of whom are very talkative, preachy, and intelligent. It might be that finger beams are an innate power of highly intelligent people (Though there are characters in fangames that go against this.) Other powers and abilities seem to be situational based on characters, and it's hard to say what caused them.
Although it is a fangame (albeit, still canonical), in Alex's flashback during LISA: The Pointless, a character makes an offhand comment, "Finger beams? Some of these fights, I tell you... Chinese fucking voodoo." Keep in mind that this flashback takes place before the White Flash, meaning that powers weren't caused by the Flash itself. Since this takes place in a martial arts tournament, we can also assume that finger beams are an ability that can be learned. However, all this is still unofficial.