Active describes the period where attacks can hit. The hit area of the fist or foot or sword you see on-screen is handled behind the scenes as a dangerous red shape swinging around: the hitbox. For most moves, this active hitbox period persists for just a few frames, as few as two (a thirtieth of a second) for a quick jab. Most moves hover between three and seven active frames. Longer than that starts to become noticeable even to the naked eye, especially for air moves. Attacks that put an active threat out there for 10 or 20 frames (a tenth of a second to a third of a second) are almost always great staple attacks, good for many situations and easy to connect with.
This also succinctly explains why projectile attacks are good in fighting games; they’re attacks with enormous active periods basically, with a projectile that lasts two seconds basically being an extra hitbox for 120 frames. See also framedata.