Bleeding is a damage over time state that is applied by various special attacks by many different combat professions. In addition to the initial regular damage, this state incurs bleeding damage on the target of the debuff. The bleeding damage is much lower than the normal weapon damage, but over time it can be significant--especially if you are able to attack the opponent for a period of time. Bleeding causes damage every 20 seconds, and is listed in the combat chat window as an orange line. Fire damage is also listed in orange.
A Doctor with Pathology I can reduce Bleeding by using the /bandage command, though it may take several tries for powerful bleed attacks. Improved Bandage, which is learned at Pathology III is more effective at reducing Bleeding. Blood Chowder and Veronian Berry Wine are foods that can reduce the chance of Bleeding, as well as absorb some of the Bleeding damage if it does stick.
Before the Combat Upgrade, some special weapons also inflicted bleeds without having to execute a certain attack. All damage over time weapons have been removed from the game, with the exception of Commando heavy weapons, which have abilities (some of the DOT) built in.