For the Imperial legion the basic organization is like this:
1 Legion = 10 Cohorts
1 Cohort = 6 Centuries
1 Centuria = 10 Contubernia
1 Contubernium = 8 men
As well as the foot soldiers there would be a cavalry unit called an alae, usually recruited from non-Roman allies they would still have a Roman commander, a decurion. Indicative of the low regard the Romans had for cavalry, this unit would often number no more than 300 men. Also there could be auxiliary units of colonial troops drawn from the provinces to back up the legions. An auxiliary unit often consisted of horsemen (for the alea), archers and slingers. Auxiliaries were used for 2 things: helping legions and, most of all, defending the frontiers. The soldiers of these auxiliaries, called auxiliarii, were never Romans. They were always employed far away from their birthplace, for understandable reasons (You're not likely to start a rebellion several thousand miles from home surrounded by strangers). The auxiliary soldiers, who drew less than half the pay of the legionaries, were granted citizenship for themselves and for their kids on their discharge after 25 years. The sons of an ex-auxiliarius almost always served the Roman army as a legionarius. When an auxiliarius got the civil rights, he often got the name of the reigning emperor.
A legion consisted of around 4,800 men-at-arms plus auxiliaries and a large number of camp followers, servants and slaves. At its disposal were about 60 catapultae and ballistae. A catapult threw rocks, and a ballista shot arrows. The campfollowers would be doctors, bakers, executioners, veterinary surgeons, writers, carpenters and blacksmiths. The legion could provide for itself this way and could thus contain as many as 6,000 fighting men although at times in Roman history the number was reduced to 1,000 to curb the power of mutinous commanders, Julius Caesar's legions had only around 3,500 men.
Each legion had a standard which was a pole with the figure of an eagle at the top, made of gold, this was called an eagle. In addition a legion carried standards holding a portrait of the emperor, special flags, and possibly a legion symbol. These standards were objects of worship in the official religion of Rome. According to the Roman historian Suetonius, when Cestius Gallus and the Twelfth Legion where defeated, the Judaeans captured their eagle an act of utter humiliation.
The legion would be commanded by a legate or "legatus". Aged around thirty he would usually be a senator on a three to four year appointment by the emperor. Some senators became very good generals and served much longer (Julius Caesar for example). In a province with only one legion, the legatus also served as the governor. In provinces with multiple legions, each legion had a legatus and the provincial governor had command of all of them. Immediately subordinate to the legate would be six tribunes, one would be the second in command of the legion and would be a noble heading for the Senate usually without much combat experience. The five other tribunes would be staff officers and these men would have much more experience than the higher-ranking senatorial tribune, having just served about three years as independent commanders of auxiliary cohorts. There would also be a group of officers for the medical staff, the engineers, record-keepers and the praefectus castrorum (commander of the camp) as well as other specialists such as priests and musicians.
There are six centurions in the cohorts numbered 2 through 10, and presumably the cohort as a whole was commanded by the centurion with the highest seniority. The first cohort consisted of 5 centuries instead of 6, but with each centuria the strength of a maniple (2 * 80 men). The first cohort, with the double-sized centuries, had five centurions, and these outranked all the other centurions in the legion. The centurion of the highest centuria (the first centuria of the first cohort) was the centurio primus pilus.
Each centuria would be commanded by an officer called a centurion, to assist each centurion there would be an optio, a soldier who could read and write. The title optio ad spem ordinis was given to an optio who had been accepted for promotion to the centurionate, but who was waiting for a vacancy. A skilled centurion had a good chance to become commander of a camp (praefectus castrorum). Every centuria also had his own fieldsigncarrier, a signifer.
Many centurions rose trough the ranks by merit but also trough connections, which were very important in the Roman world. Some centurions, however, were directly appointed by provincial governors from members of the wealthy classes. These were 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. They would become the commanders of the high ranked centuria. While the legatus and tribunes were political appointees, assuring that the army was controlled by men trusted by the Emperor, most of the centurions were skilled professional soldiers who would be relied on to run a legion on campaign and in battle.
A contubernium consisted of eight men called miles ("soldier") or legionarius ("legionary") in Latin. The Roman soldier was a Roman citizen under 45 years of age. The soldier enlisted for twenty years of service, a change from early practice of enlisting only for the duration of a campaign. The eight men of a contubernium shared a cookpot and a tent. Such a tent wasn't very spacious: about 4 by 3 meters, with a tiny front room. Each contubernium had a mule, to carry the heavy stuff when the army was marching. The soldiers were trained especially hard, discipline was the base of the army's success and the soldiers were relentlessly and constantly trained both with weapons but especially with drill. Forced marches with full load and in tight formation were trained on a regular basis. And infractions were heavily punished by the centurions (hence the executioners).
I can highly recommend the book: "The conquest of Gaul" by Julius Caesar for a great description of the legions in action.