In Rome there was no separation of church and state. The second king of Rome Numa Pompilius created Rome's main religious institutions and organized all the religious ceremonies himself. The priesthood was usually held by members of the upper class holding government functions. It would be like the President or a high ranked government official today holding the post of Pope or something equivalent! For example, Julius Caesar was Pontifex Maximus which was the highest position in the ancient Roman religion. During the Empire, there were public vows made to the emperor himself.
(The Ancient City - Connolly, Hazel)
The Roman religion was also mostly based on the respect of rituals and traditions rather than based on faith. It was about respecting the way of the ancestors mos maiorum and giving to the gods through prayer, rituals and sacrifice. By giving to the gods through sacrifice, a Roman expected to get in return the favor of the gods. This was the principle of the do ut des. Prayer was also very important. A sacrifice could not be performed without prayer. There were exact words that had to be read and various prayers depending on the deity invoked. Even private prayers at home had to be recited word for word. Rituals also had to be enacted very accurately. If a mistake was made, the ritual or the whole festival would have to be started over.
Temples were built at sacred places or they were built to commemorate an event or a victory (and to thank a specific god for that victory). They had a main room (a cella) with an image or a statue of the god to whom the temple was dedicated. There was a small altar for incense or libations (the pouring of liquids such as olive oil). There could be artworks or treasures (gold) plundered in wars and dedicated to the particular god. There was also a bigger altar outside used for animal sacrifice.
Religion was part of daily life for the average Roman citizen. Every home had a shrine in front of which the family would pray or make offerings or libations. There were also neighborhood shrines where people would go to pray and there were sacred places in the city and in the countryside. The Roman calendar had many religious holidays: there were about 40 religious festivals per year, with the festivals lasting sometimes many days! One such festival was the Compitalia which was in honor of the Lares Compitales which were household deities of the crossroads (compitum in latin means a crossroad). Each household would place a statue of the goddess Mania in front of their door, they would hang human figures made of wool at their door (so as to satisfy the goddess and the Lares) and they would offer sacrifices at places where two roads meet. They sacrifices often consisted of honey cakes.
There were many types of priests. The augurs practiced augury meaning that they read the will of the gods. For example, they would study how birds fly (whether they flew in group or separated), what noises they make and interpret the will of the gods. The augurs would influence regular Romans or public officials even on matters of war and commerce. The pontiff (pontifex) was a member of the most famous colleges of priests called the College of Pontiffs.
Women also participated in separate religious activities and some rituals could only be performed by women. The Vestal Virgins was a priesthood composed of six women which lasted for centuries. These women were devoted to the goddess Vesta, the goodess of the hearth of the Roman state and its vital flame. A Vestal had a high public status, privileges and an important political influence. Unlike male priests, Vestals did not have to marry or have children and they actually had to remain virgin for life. A Vestal who lost her virginity would loose her sacred status and... would be buried alive.
Since it was about respecting so many traditions, the members of the upper class saw religion as a source of social order. The Roman religion was also what united Romans and gave them their Roman identity.
Interesting differences between the Roman religion and religion today