Other forms: involved; involving; involves
To involve means to connect. When you get involved in the Spanish Club, you attend meetings and events. If doing math well is involved in understanding advanced chemistry, that means it's a necessary part of it.
When someone is very involved, it means they are engaged or connected to a lot of activities. When someone is self-involved, all they connect with is him or herself. If getting ice cream involves walking a mile to the ice cream shop, the exercise offsets the indulgence. When no exercise is involved, it's less healthy.