For the Imperial legion, say from Julius Caesar onwards, the basic organization is like this:
8 men=1 contubernium (mess unit/tentful)
10 contubernia=1 centuria (century), commanded by the centurion
6 centuriae=1 cohors (cohort), probably commanded by its senior centurion
10 cohortes=1 legio (legion), commanded by the legatus
During the Republic the centuries had been paired
into maniples, each with a "front" and "rear" century. These terms survive
into later times, and that leads into all kinds of arguments about how
the units were arranged in battle. But in the Empire, the maniple is apparently
no longer used, the cohort being the basic tactical unit.
By the Empire all the troops are armed and armored pretty much the same way (we think!), and the century is a standard 80 men. In the mid-first century AD most legions changed their first cohort to 5 double-sized centuries, so it had 800 men but only 5 centurions, for a full legion strength of 5120. (But remember that units are ALWAYS under strength!)
The Legatus was a senator in his 30s who had been a senatorial tribune and then gone through the civilian government posts in Rome. He was appointed by the emperor and held command for 3 or 4 years, although some became very good generals and served much longer. In a province with only one legion, the legatus also serves as governor; in provinces with multiple legions, each legion has a legatus and provincial governor has command of all of them.
Second in command of the legion was the tribunus-laticlavus or senatorial tribune, a fresh-faced young man on his first job away from home. He probably relied heavily on the next man down, the preaefectus castrorum or camp prefect, a grizzled veteran who had been promoted up through the centurionate. Then came the 5 tribuni angusticlavi or equestrian tribunes, appointed from the wealthy class (just below senators). These guys actually had more experience than the higher-ranking senatorial tribune, having just served about 3 years as independent commanders of auxiliary cohorts. (An auxiliary cohort was 6 centuries of men enlisted from the provinces, and some of them were pretty barbaric. I wonder if they ever ate their commanders? It used to be said that the tribunes just held administrative posts and did not actually lead troops, but now we think that each equestrian tribune commanded 2 cohorts of legionaries.
Then come the centurions, 59 or 60 to a legion. They have their own very confusing hierarchy: There are 6 distinct steps of seniority in each cohort, from lowest to highest: hastatus posterior, hastatus prior, princeps posterior, princeps prior, pilus posterior, pilus prior. (Note that "pilus" means "file", NOT the same word as "pilum". In the Republic the triarii were sometimes referred to as "pilani".) The cohorts themselves are ranked from the First (highest) to the Tenth (lowest). In theory a centurion would start in the lowest spot in the Tenth cohort, rise to the top of that, then move to the lowest spot in the Ninth cohort, etc. Probably it never really happened that slowly. The centurions of the first cohort were called the Primi Ordines, and were headed by the Primus Pilus (which means "first FILE"!), the senior centurion in the whole legion. From there a man could rise to Praefectus Castrorum, third in command of the whole legion, and after a year in that post he'd retire in fabulous wealth and glory.
Many centurions, probably most (and probably the best), rose from the ranks by merit (and connections, very important in the Roman word!). Some centurions, however, were directly appointed by provincial governors from members of the equestrian class. These were wealthy men who decided to join the army as centurions in order to gain advantages and status, and they were apparently "fast-tracked" for promotion, rising quickly to the highest ranks over the heads of the men of lower social status who had risen from the ranks.
At the bottom end of the scale are the milites gregarii, the common soldiers. They spent a couple month in boot camp, then get posted to a legion and spend their 6 months rated as recruits. The first step up a soldier could make was to immunis, meaning he was posted to more more skilled task (clerical, craftsman, etc.) and was generally "immune" from the usual hard labor and dirty jobs such as road building. The first real promotion was to pay-and-a-half (sesquiplicarius), such as the tesserarius (guard sergeant), cornicen (horn player), etc. Next come the double-pay posts (duplicarius, optio (second in command of the century), signifer (or is he a sesquiplicarius?), and aquilifer (the Legion's eagle-bearer, a VERY prestigious post!). Then, hopefully, would come the big step up to centurion.