Century comes from centum, a hundred, meaning a unit of hundred individuals (whence also our 'century' consisting of 100 years). Originally it was a unit for the division of the Roman Citizenship (ascribed to the Roman king Servius Tullius, introduced probably somewhere in the 6th century [ha!], which is why you'll often find this mentioned as 'Servian constitution'), where the citizens were divided into different classes according to their wealth. This was primarily a military-oriented division (different wealth meant different available equipment since the soldiers had to pay for it, meaning the armed citizenry was spanning the spectrum from fully-armed and armoured hoplites and knights over javelin-armed skirmishers down to slingers and stone-throwers).

60 such centuries made up the early Roman legion of 6000 men (earlier 3000 infantry and 300 cavalry). This was also used as a political sub-unit used for voting in the comitia centuriata assembly, where each century gave its vote as one. However, especially in the later comitia, there were more centuries allotted to the higher wealth classes than to the lower, and since the number of centuries wasn't modeled to the total number of citizens, membership in each such century varied wildly (the wealthier ones contained less people, which proportionally gave the rich more voting power and gave heavy importance to their vote).

In the military, theoretical strenght is almost never reached - there are always soldiers sick, on leave, detached to other units or on special duties, there are losses that aren't yet replaced and so on. In any case, while the name of the sub-unit (century) stuck (think of how different a division, bataillon or brigade can be in manpower - incidentally, the German police forces also divide their tactical units into Hundertschaften, literally 'group of a hundred', which usually contains 117 men), the size changed based on general strategy or preferences. In the legion of the 3rd century BC, a century on paper had 60 men, with two centuries making up a maniple of 120 men, while in the later republic, six centuries of 80 men made up a cohort.

A Roman legion was divided into ten cohorts. Each cohort was divided into six centuries (or centuriae) of eighty men. Each century was further divided into ten groups of eight, known as a contubernium, who shared a tent. In other words, a cohort would have 480 men.

However, the first cohort of each legion only had five centuries, but each of these were double strength, meaning 160 men rather than 80, for a total of 800 soldiers.