Edge-hogging is the act of holding onto a ledge so that a recovering opponent cannot grab it. It is a form of edge-guarding. Doing so earns one the "Edgehogger" bonus.
The most common method of edgehogging in competitive play is to wavedash back while near the edge, which should result with the characters sliding off of the stage, falling a short distance, and then quickly grabbing the edge. The character then usually uses the invincibility from the edge to avoid being hit by the opponent's recovery move. Other methods include moonwalking, doing a reverse neutral B then sliding off, using certain attacks which allow one to grab the edge faster or while attacking (i.e. Bowser's Whirling Fortress), or simply short hopping back and grabbing the edge.
Another method, called the Pivot Edge-hog, involves pivoting very close to the ledge. The character will skid a very short distance during the turn-around animation and land on the ledge. Certain characters, such as Fox from the original Super Smash Bros, can do something similar with the turn-around animation out of a dash.