Living weapon is a keyword found on Equipment. When that Equipment enters the battlefield, you create a 0/0 black Phyrexian Germ creature token and attach the Equipment to it immediately. The Equipment's bonuses keep the Germ alive—a +X/+X boost makes the 0/0 into a functional attacker or blocker.
The key nuance: the Germ only exists because of the Equipment. If the Equipment becomes unattached (you move it to another creature, or it stops boosting the Germ to positive toughness), the Germ dies as a 0/0 state-based action. The Equipment itself stays on the battlefield—it doesn't die with the Germ, it just becomes unattached.
In Commander, living weapon gives you a body and equipment in one card, dodging the usual "you need a creature first" problem. Watch out for equip costs and removal that targets the Equipment versus the Germ separately.