common-law
Customary law (in English derived common law legal systems) is the "long established" way that issues that arise in a lawsuit were treated by people in the community is which the issues arose in a matter which is not the subject of any controlling statutory or case law, that a court gives the force of law in that lawsuit.
Often customary law is derived from tribal law, village customs, or practices that had previously been handled by clergy. This was especially a major source of law when the common law was just emerging, or in colonies (e.g. in Africa or India) with a pre-existing indigenous population.
For example, suppose that there is a dispute over whether an engagement ring must be returned when an engagement is broken, among people who live in a Welsh language community in Wales where the long standing tradition and social norm is that when an engagement is broken, the prospective wife may keep the engagement ring as a sort of compensation for the shame and reduction in their marriage prospects of having endured a broken engagement. A court presented with that situation with no statutes or case law that control its decision might adopt that rule as a rule of customary law.
Customary law also applies to customary commercial practices in novel new commercial contexts to supply default terms in a transaction when no term in a contract specifies the relevant term of the agreement.
For example, suppose that in Facebook Marketplace transactions, it is customary for buyers to come to a place designated by the seller, but no one specifies where the goods are to be picked up, and the buyer and seller are very far apart so that the shipping costs would be considerable. A judge might apply commercial practice and custom to hold that any cost of shipping the goods from the seller to the buyer in the absence of an agreement in a Facebook Marketplace transaction are at the expense of the buyer.
The traditional legal doctrine of "livery of seisin" which held that a real estate transfer was final when a physical object from the property being transferred was handed from the seller to the buyer (or transferor to transferee in the case of a gift of real estate), for example, started as customary law, became case law, and then was repealed by statutes which provided that title to real estate could be transferred only with a signed deed.
Ultimately, almost all common law case law rules derive at some point, often in the remote past, from customary law. But, over time, the number of potential legal issues which are not controlled by statutes or case law grows smaller and smaller as customary law rules are converted to case law, or are superseded by comprehensive statutes that address almost all issues that can arise in a particular kind of legal dispute.
Say you're in England in 1750 and you have a dispute over whether
someone actually offered to sell you a certain slave at a specified
price. He claims he was drunk and only joking; you claim he did make
the offer while expecting that you would be unable to pay that much,
and when he found you with the money and attempting to pay him he
tried to back out. The case goes before a judge.
No statutory law of England said there was a right to own slaves, but
it was customary that judges treated slaves in some important ways
just like other chattels. (Until 1772.)
Generally, a point like this would arise from statements and assumptions, either explicit or implicit in previous case law decisions. Slavery was showing up in case law in England from the very earliest cases in the formative period of the common law legal system, perhaps in the 13th to 15th centuries. Customary law generally concerns matters that have no foundation explicit or implicit, in previously decided precedent making case law.
A better example might concern the question of when a wild animal becomes a pet who becomes the property of a pet owner. If customary folk wisdom was that a wild animal ceased to be wild and became a pet who was owned when (1) it returned of its own free will to be fed by the person alleged to be a pet owner on a regular basis that (2) the alleged pet owner put out with an intent that this particular animal be fed, then a court might adopt this customary folk wisdom as customary law regarding when a wild animal becomes property. In future cases after the decision adopting this rule of customary law, however, this would then be case law rather than customary law.
Rules of law established by case law are also known as common law rules of law.