The expanding Amberverse from dkap-amber
Technically, an Old Lord is any being who successfully pulls themselves ashore from the Sea of Chance onto the Courts of Chaos. A while ago, to maintain the quality, rather than just the luck (they might have been just created, right off shore, and didn’t have any real strength to survive the sea) whenever there is such an event, all available current Old Lords attack them en masse, with the express hope that the individual so pulled is worthy of the rank.
Originally the Old Lords were those that were on the Chaos Promontory, when the last (or several of the last) Sea of Chance waves washed over, obliterating everything. Those that got off, were safe, and got back once the wave receeded were considered Old Lords. Some have been known to do this several times over. Since sometimes it is hard to tell time in the Sea, an Old Lord could, theoretically, just be getting around to coming ashore at any particular point in time. Hence the techicality above.
All of this is opposed to being a Demon, which is defined as someone else doing the pulling, or rescuing from the Sea of Chance or the Abyss, or descendant from one such. Pulling someone else out of the Sea grants the puller the chance to name said Demon.
It is said that there is a “look” about Old Lords. Although that might just be the inherent paranoia and trauma left behind after their “welcoming party” tried to annihilate them.