Ok, so this is one of the stupidest names for something in football. Most offensive formations reflect in some way, some sort of reality (i-formation, wish bone, spread). Wildcat? What on earth is that supposed to mean!
Well, it is a bit ‘wild’ but it’s not that crazy. The general idea is that the quarterback is not on the field or is in a wide receiver slot. Instead of taking the snap from center, a running back or another back takes the snap. That makes an immediate run threat the defense has to be ready for. Essentially, it makes a typical offense of 11 guys actually have 11 guys the defense must defend instead of just 10 guys.
See below for a basic wildcat formation. The QB is generally in the huddle to help keep the defense in the dark but then lines up as a receiver while a super fast RB or WR lines up in the usual QB position and takes the snap directly from the center.