The Imperial Roman Army is a favorite topic, and its broader organization has been well-treated elsewhere. However, I haven't been able to find a really thorough treatment of its mainstay - the legion - so I thought I'd provide one. Talking about the "Roman legion" is difficult, as the unit changed a great deal throughout the history of the empire. However, most consider the 1st and 2nd CE to be the high point of the Army, so I'll aim for the Trajanic legion of the late 1st century.
The military of the Imperial period was fully professional, made up of soldiers who served for at least 16 years and often more. There were 30 consecutively-numbered legions in Trajan's time, each with its own unique character and set of symbols and accolades. A typical legion boasted a full-strength complement of 5,500 men, including legionaries, officers, and support staff. Thus, the legions as a whole constituted some 160,000 men, fully half of the Roman military. The rest was made up of the Praetorian guard, the imperial fleet, and the auxilia, or auxiliary forces.
The largest part of the legion in Trajan's day was its 5,120 legionaries, who were organized into discrete sub-units within it. The smallest unit was the contubernia, which consisted of 8 soldiers. The men of a contubernia shared a pair of rooms when the legion was housed in a permanent camp, and a single tent when on campaign. Each contubernia was allotted a pack mule, which carried the tent and other common equipment when the legion was on the move. The contuberniae were grouped into 80-man centuria, or centuries, which were the legion's smallest fighting units. These centuries were paired in camp and on the march into units called maniples; however, the maniple was primarily a throwback to an earlier organizational system and was not particularly significant tactically. Instead, the centuries were grouped into ten cohorts. Cohorts II-X consisted of 480 men in 6 centuries, while cohort I consisted of 800 men grouped in 5 double-sized centuries.
Each century was commanded by a centurion, who was assisted by an optio, or junior officer, who would take control of the century if the centurion was incapacitated or killed in battle. The legion as a whole was commanded by the legatus legionis; this was a post given to members of the senatorial class as part of their political careers, so the quality of the legati varied widely. Beneath the legatus were six military tribunes, who were responsible for judicial and administrative duties. These officials were assisted and advised by the praefectus castrorum, the camp prefect, who ranked just below the senior tribune, and who would assume overall command of the legion if the senior tribune and legatus were absent. This prefect was normally a life-long soldier who had worked his way up through the centurionate and commanded the first cohort.
Each century had a tesserarius, who was in charge of guard and picket duties, a cornicen or horn-blower, and a signifer, who was responsible for the century's battle standard and kept track of pay and expenses. In addition, the legion as a whole had an aquilifer, who carried the legion's aquila (eagle) and imago (likeness of the emperor).
Auxilia units were deployed with the legion to support the legionary core by providing light skirmishers and cavalry. There were three auxilia unit types, and each came in two sizes: quingenaria and milliaria. These types were, in order of importance and prestige, the ala, cohors peditata, and cohors equitata. Alae were combat cavalry, divided into turmae, or squadrons. A squadron consisted of 30 cavalrymen, a commanding decurion, and an assistant officer called a duplicarius. An ala quingenaria consisted of 16 turmae, while an ala milliaria consisted of 24 turmae. A cohors peditata was an infantry unit made up of standard 80-man centuries; the quingenaria type was based on the standard legionary cohort, while the milliaria was based on the cohort I type. This type of unit was commanded by a praefectus cohortis and had the same officers and specialists as a legionary cohort. Finally, the cohors equitata was a mixed unit - a cohors equitata quingenaria had 4 turmae and 6 centuries, while a cohors equitata milliaria had 8 turmae and 10 centuries.
The legion's soldiers were extremely well protected. Each soldier wore a type of cuirass called the lorica segmentata, and a helmet called a galea which by Trajan's time had evolved to include neck and cheek guards. Leg armor was sacrificed for the sake of mobility, but the legionaries could easily protect their legs with the shield they carried, the scutum. This was a large, rectangular shield made of layered plywood and faced with leather; it had a metal boss in the center which protected the grip, as well as metal-rimmed edges and gilded or silvered decorations around the boss, which represented the thunderbolts of Jupiter.
The soldiers were also well armed. Each legionary carried a short stabbing sword called a gladius, supplemented by a pugio (dagger), and two pila (javelins.) A legionary could throw a pilum more than 60', and the weapon was designed to bend or break on impact, making it impossible for an enemy to turn it around and use it against the legion.
A legion deployed for battle in two lines - the first, from right to left, consisted of cohorts I-V, while the second, also right to left, was made up of cohorts VI-X. Each cohort was arranged with its three maniples side-by-side; the centuries within the maniples were deployed one behind the other, so the cohort arrayed for battle was three centuries wide and two deep. The first cohort would deploy in a square formation two centuries on a side, with the extra either backing up the other four or held in reserve. The best soldiers were usually assigned to cohorts I, III, V, VI, VIII, and X, meaning the center and ends of each line were the strongest. The cohorts were drawn up with a cohort's width between them, and the two lines were staggered so the cohorts of the second line overlapped the gaps in the first. Finally, any auxilia or numeri - specialists or irregulars - attached to the legion were deployed as skirmishers. Once the battle began in earnest, these skirmishers retreated through the gaps in the lines, which were then closed by the expedient of moving the rear century of each maniple up onto the battle line.
Roman doctrine included many different strategies for a legion engaged with the enemy, which took into account terrain, relative strengths of the two forces, and the strengths within the Roman line. The strategies were all aimed at turning the enemy's line, thus forcing them to fight the legionaries on more than one front, and more than likely breaking their formation to pieces. This well-developed doctrine, as well as the superior equipment and training of the legionaries, made the Imperial legion the mightiest fighting force of the ancient world.
I got most of the info for the essay from which this is condensed from Julian Bennett's Trajan, Gary Brueggeman's The Roman Army, Henry Parker's The Roman Legions, George Watson's The Roman Soldier, and Graham Webster's The Roman Imperial Army. All are highly recommended.