A sign of acceptance, approval or encouragement, made with closed fingers and the thumb extended upwards.
It is widely known that this gesture originates from the gladiatorial contests of ancient Rome, in which the fate of a losing fighter was decided by gestures from the crowd. Okay, so if it's widely known, why does it need to be included here? Well, as so often with etymology, the truth isn't quite so simple.
The belief that the 'thumbs-up' and 'thumbs-down' gestures indicated approval and disapproval respectively entered the public consciousness with Jean-Léon Gérôme's 1872 painting 'Pollice Verso'. The 'thumbs down' gestures of the crowd in Gérôme's popular picture were interpreted by the 19th century public as signs of disapproval. Actually, the artist probably never intended that, as 'pollice verso' just means turned thumb.
Prior to that date the references in print don't support that interpretation either. The earliest such citation in English is Pliny's Natural Historie, translated into English in 1601 by Philemon Holland:
"To bend or bow downe the thumbes when wee give assent unto a thing, or doe favour any person."
There's now some debate amongst scholars as to the meaning of the thumb gesture in Roman amphitheaters. The meaning of the original Latin texts is difficult to interpret. Some say that Holland mistranslated Pliny's original 'pollices premere' text and that it should be 'to press the thumbs' rather than 'bend the thumbs'. Two positions are argued:
One view is that we just have thumbs up and down the wrong way round.
The other camp say that approval was indicated by a closed fist and disapproval by showing the thumb (either up or down).
Either way, a defeated gladiator in the Roman Coliseum looking toward the crowd for support, would have hoped not to see any 'thumbs up'.