We know that the ancient Greeks had a massively entertaining sets of gods and goddesses. So it's no wonder that when Rome conquered Greece, they replaced their own dull pantheon with renamed versions of Zeus, Athena, and the others. But not all Roman gods were Greek copies — here are a few of the more important ones.
Despite Rome's wholesale acquisition of the Greek pantheon, perhaps their most important deity was uniquely their own. Janus was the god of beginnings and endings, transitions, doors, basically anything that had two different but complementary functions — epitomized by the fact he had two faces. Janus was so important he remained a part of the Roman pantheon even after they had co-opted the Greek gods, and arguably stayed their most important deity, receiving the bulk of the worship for important beginnings and ending like births, marriages, days, seasons, deaths, etc. Since he presides over beginnings, he had to be worshipped first in all major religious ceremonies; this also made him the god of omens, and the Romans loved omens and trying to divine the future. Even though Jupiter was the king of the gods after the great Greek pantheon acquisition, Janus was the door between heaven and earth, and had been since the beginning of time — which effectively meant Janus could have metaphorically locked Jupiter in his room. He didn't, of course, but he could have.
When Rome was founded, it was near a group of people called the Sabines. Some of the Sabines were absorbed into Rome peacefully — some were very much not — but they brought Quirinus with them, a god of war, before the Romans took Ares and redubbed him Mars. By the first century B.C., somehow Quirinus had become the divine form of Rome's legendary founder Romulus, making him a pretty big deal. He effectively became the god of the Roman state; as the republic turned into an empire, and the empire grew, he was eventually supplanted by Jupiter and the other formerly Greek gods.
While Rome obviously loved borrowing religion from their Greek neighbors, Mithra was one of the few divinities that came to the empire from somewhere else. Original a Persian god, Mithra began to be worshipped by soldiers in the Imperial armies, but as part of a cult — they had secret rituals and prayers and temples, although it doesn't seem like there were any crazy cult leaders or nefarious dealings. They just worshipped Mithra and hung out and refused to tell other people about it, almost like a club.
Not all gods needed worship from the masses; every Roman household had a variety of smaller, more personal gods that they worshipped at a family altar right in their home. Among the most important were the Lares, the gods of the family, which included the spirits of their ancestors. Obviously, as part of the family themselves, they had a vested interest in keeping their descendents happy and protected, and the descendents had good reason to keep them satisfied. They were offers of small tokens of food and grain and wines every day, but more elaborate rituals were held for important family events like weddings and births.
5) Dis Pater
Originally a god of land-based riches — which is to say fertile land, precious minerals, etc. — Dis Pater eventually turned into a god of the underworld because anything precious in the ground was considered the treasure of the underworld. It's kind of a weird association to move from God of Gems and Stuff to God of the Place People Go When They Die, which is likely why he was absorbed into Pluto, nee Hades, when the Greek gods came over. Pluto was a much more impressive, imposing god of the underworld.
The Etruscan were a group of people living near Rome that got absorbed/conquered early in Rome's history. As such, Orcus is believed to have been the Etruscan god of death and the afterlife, alongside Rome's Dis Pater. Eventually, Orcus came to represent the more evil, monstrous aspects of the afterlife, and became seen as a demonic tormentor of the damned; between that distinction and the fact that he was mostly worshipped on farms instead of in the cities, he continued to exist as a divinity long after Dis Pater had disappeared into Pluto. Fun fact: Orcus is the origin of the name of J.R.R. Tolkien's evil Orcs.
The origin of Liber is mostly unknown — he was definitely worshipped in Italy for quite some time, and was joined by his female companion Libera — but he was a major god in the Roman pantheon. He was the god of male fertility, the transition to adulthood, the bounty of harvest and the common man — a weird conflation of rights, manhood, fertility, and partying. Perhaps "liberty" is the best word for it. Rome devoted a whole month to celebrating him — the Liberalia festival — in which of course a giant penis was paraded about, ostensibly to protect the crops, as penises are wont to do. Eventually Liber was superceded by the Greek god of wine, Dionysis, renamed Bacchus, but worship of Liber continued, evolving into cults with outrageously unsavory reputations for sexual free-for-alls and occasional murder.
8) Divus Julius
Going into all the reasons why Julius Caesar was deified while he was still ruling Rome would be incredibly complicated, but you need to know two things: 1) Caesar had been a friend to the common people of Rome forever, and they loved him even before he brought an end to a horrible civil war and ruled the nation, and 2) Caesar's willingness to be honored, including having a month named after him, a priest devoted to him, countless statues, etc., was a primary reason why many people in the Senate wanted him dead. But after the murder, Marc Antony helped get Julis officially recognized as a deity, which gave him more or less a divine mandate to punish Caesar's murderers and take over, although that also got super-complicated thanks to Julius' nephew Octavian. Julius Caesar's divinity was primarily a political move, first by him, and continued by his successors to legitimize their power.
9) Divus Augustus
Julius' nephew Octavian did not make the same mistakes his uncle had. Even when he defeated Mark Antony and became emperor, he refused to be voted divine honors while he was alive — and since he ruled for over 40 years, that was quite a long time. Even when the "savages" of Rome's furthest provinces specifically requested to be allowed to worship their Emperor as a god, Augustus did not want to follow in his uncle's footsteps. But he also didn't want to offend his new subjects. As such, he basically told them to worship the spirit of Rome itself, and hey, if they happened to mention him too, no biggie. But he refused to allow any pretense of divinity in the more traditional Roman portion of the empire. He also helped restore countless temples and festivals, thus worshipping every god he felt he should; between that, and the fact that he ruled for 40 pretty peaceful, incredibly prosperous years, when he died in 14 AD both the common people and the senate were more than happy to declare him an official god.