The most common use of the term is in the case of English peerage dignities. Most such peerages pass to heirs-male. But the ancient baronies by writ, and possibly some very old earldoms, pass instead to heirs-general. If such a peerage is held by a man whose heir-at-law is neither a male, nor a woman who is an only child, on his death it goes into abeyance between two or more sisters or their heirs, and is held by no one till the abeyance is terminated. If eventually only one person represents the claims of all the sisters, he or she can claim the termination of the abeyance as a matter of right.
It is entirely possible for a peerage to remain in abeyence for centuries. For example, the baronies of Burgh and Strabolgi, which had a common descent, were in abeyance for 547 years, between 1369 and 1916.
The crown can also call the peerage out of abeyance at any moment, on petition, in favour of any one of the sisters or their heirs between whom it is in abeyance.
The question whether ancient earldoms created in favour of a man and his heirs go into abeyance like baronies by writ has been raised by the claim to the earldom of Norfolk created in 1312, discussed before the Committee for Privileges in 1906.
It is common, but incorrect, to speak of peerage dignities which are dormant (i.e. unclaimed) as being in abeyance.
Adapted from an old 1911 Encyclopedia