Terminus post quem
Terminus post quem ("limit after which", often abbreviated to TPQ) and terminus ante quem ("limit before which") specify the known limits of dating for events. A terminus post quem is the earliest time the event may have happened, and a terminus ante quem is the latest. An event may well have both a terminus post quem and a terminus ante quem, in which case the limits of the possible range of dates are known at both ends, but many events have just one or the other.
Similarly, terminus ad quem ("limit to which") is the latest possible date of a non-punctual event (period, era, etc.), while terminus a quo ("limit from which") is the earliest.
For example, consider an archaeological find of a burial that contains coins dating to 1588, 1595 and others less securely dated to 1590-1625. The terminus post quem for the burial would be the latest date established with certainty: in this case, 1595, based on the latest securely dated coin. A secure dating of another coin to a later date would shift the terminus post quem.
An archaeological example of a terminus ante quem would be deposits formed before or beneath a historically dateable event, such as a building foundation partly demolished to make way for the city wall known to be built in 650. It may have been demolished in 650, 649 or an unspecified time before; all that can be said from the evidence is that it happened before that event.
Other examples of things that may establish a terminus are known dates of death or travel by persons involved, a particular form of heraldry that can be dated (see pastiglia for example), references to reigning monarchs or office-holders, or a placing relative to any other events whose date is securely known.
A terminus ante quem non differs from a terminus post quem by not implying the event necessarily took place. 'Event E happened after time T' implies E occurred, whereas 'event E did not happen before time T' leaves open the possibility that E never occurred at all.