The minute, in this case, is a unit of time. As a unit of time, the minute is equal to 1⁄60 of an hour or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second which would result in a 59-second minute, but this has never happened in more than four decades under this system).
By contrast to the hour, the minute (and the second) don't have a clear historical background. The only thing traceable, is that it only started being recorded in the Middle Ages, due to the ability of construction of "precision" timepieces (mechanical and water clocks). However, no consistent records of the origin for the division as 1⁄60 part of the hour (and the second 1⁄60 of the minute) have ever been found, despite many speculations.
Historically, the word 'minute' comes from the Latin pars minuta prima, meaning "first small part". This division of the hour can be further refined with a "second small part" (Latin: parte minutae secundae) and this is where the word 'second' comes from. For even further refinement, the term 'third' (1⁄60 of a second) remains in some languages, for example Polish (tercja) and Turkish (salise), although most modern usage subdivides seconds by using decimals. The symbol notation of the prime for minutes and double prime for seconds can be seen as indicating the first and second cut of the hour (similar to how the foot is the first cut of the yard or perhaps chain, with inches as the second cut). In 1267, the medieval scientist Roger Bacon (Mhmm, bacon.. yummy...), writing in Latin, defined the division of time between full moons as a number of hours, minutes, seconds, thirds, and fourths (horae, minuta, secunda, tertia, and quarta) after noon on specified calendar dates.