What was the political system of the peoples living in what is presently the Russian state during the time of the Roman empire? Also, was there any communication between these peoples and the Roman empire? I know its a wide time period but if someone could shine light on this topic that would be great.
I'm going to assume that you are referring to the classical Roman empire, as it is rare for someone to refer to the other two TED mentioned without someone explicitly adding Byzantine or Holy to modify it. Essentially, at the time of the classical Roman empire there wasn't a Russia to speak of at all. Here is a map showing what Greco-Roman geographers referred to the rest of the world. The area where Russia would develop was lumped in with the Iranian peoples known as the Sarmatians. It is rather clear that Rome had little contact with the people in the region.
There isn't much that is known of the region until the eastern Slavs had both settled the region and developed a written language in around the end of the 9th century. Much of the first millennium AD simply saw various migrations through the area and a series of imperial encroachments in the land which would become Russia. As zefciu noted, its not until the 14th century that the modern Russian state developed with Moscow as its capital. However, it is possible to trace the decent back to the Kieven Rus, the precursor to Russia,
The establishment of the first East Slavic states in the 9th century coincided with the arrival of Varangians, the traders, warriors and settlers from the Baltic Sea region. Primarily they were Vikings of Scandinavian origin, who ventured along the waterways extending from the eastern Baltic to the Black and Caspian Seas. According to the Primary Chronicle, a Varangian from Rus' people, named Rurik, was elected ruler of Novgorod in 862. In 882 his successor Oleg, ventured south and conquered Kiev, which had been previously paying tribute to the Khazars; so the state of Kievan Rus' started. Oleg, Rurik's son Igor and Igor's son Sviatoslav subsequently subdued all local East Slavic tribes to Kievan rule, destroyed the Khazar khaganate and launched several military expeditions to Byzantium and Persia.
To sum, there was no unified state in the region which would become Russia in the time of the classical Roman empire (or even during the height of the Byzantine empire for that matter). The governmental structure of the peoples living in that region would likely have been tribal nomadic, although it is difficult to tell due to the lack of written records. In addition, various empires claimed territory in this region for various lengths of time.
Basically, from south to north, you had pastoral Iranian tribes, then Slavs, then Finns. After a while (around the third century BCE), the easternmost German tribes (the Ostragoths) learned the pastoral lifestyle from the Iranians, and took over their territory in southern Russia. A bit later (370), Huns arrived from the east, and conquered the Ostragothic territory (along with the Ostragoths). That whole time, in the north the slavs slowly pushed the Finns north.
There was generally communications between the eastern half of the Roman Empire and whomever held the southern Russia pastoral territory at the time, although sadly much of it was via warfare. The activity in the northern part is known to us only through archeology. It wasn't until Byzantine times that any kind of regular trade was established between Constantinople and northern Russia.