Fox-trotting is the technique of moving along the ground by repeatedly executing a character's initial dash animation, as opposed to using their full dash. Since some characters' initial dash animations move them faster than their full dash, these characters are able to run faster the use of fox-trotting. Fox trotting is predominately used in Melee, and characters with which the technique is most commonly executed are Fox, Captain Falcon, Roy, and Marth.
One way of performing a fox-trot is by flicking or tapping the control stick sideways to perform an initial dash, returning the stick to a neutral position to cancel the onset of the full dash, and then tapping the stick sideways again to perform another initial dash. This sequence can be repeated to move the character along the ground, and the timing of each character's fox-trot is different. In Brawl, there is a small possibility that a character will trip during their initial dash animation, thus fox-trotting increases the risk of tripping.
In Melee, fox-trotting can also be performed by jump-canceling a dash into a wavedash, then dashing again, or by dash-canceling a full dash into another dash.
Dash Sliding, a variant of Fox-trotting, is described as a constant state of dashing. It is done by mashing the control stick in an direction, and then returning it to neutral, then repeating, similar to a Foxtrot. It differs from a foxtrot in that when doing a Foxtrot, a dash-dance was needed to change directions. In a Dash Slide, it is possible to shield cancel out of it. Also, since most characters hunch over in the animation, the player is a smaller target for an added bonus. This can be used as a mindgame, or as a speed advantage over the player's opponent.
The term was coined by Clan, Hoboking, and Legend of Brawlcentral. The whole thread is here: [1]