How did the viking's form of government differ from the other countries at the time?

This is a really interesting question, and a complex one, too. There were major shifts in the middle of the Viking Age, right around the year 900, which I'm still trying to figure out, but I'll give this a whack.

Early viking attacks around the year 800 were mostly small-scale raids—a few dozen to maybe a hundred guys, locally organized, and maybe without any major political goals in mind. But these were the decades when Charlemagne was consolidating power, the Mercians were building a superstate at the center of England, and the Danish nobility was winnowing itself down to a single family working as a ruling dynasty. These consolidating states all outgrew themselves around the year 830 and started to collapse into infighting and civil wars.

Lesser lords took advantage of this disintegrating situation. Charlemagne's descendants were duking it out for leftovers of his overgrown empire, Wessex was on the rise in England, and aspiring Norse aristocrats started setting themselves up as petty kings around the Irish Sea, most notably at Dublin.

The next generation is particularly rough in the written sources. The kings who had supported the churches and their scriptoria (libraries) were now scrambling for power, which meant that not only were their former benefactors in the church now left without an income, but also that the church treasuries were a prime target for pagan and Christian princes alike. It's hard to pick through church accounts of this period, which read as if this were the end of the world, but by the 860s, Norse-speaking warlords had established themselves on par with Christian kings in the British Isles and France. Perhaps worst of all, they were beginning to ally with the church—the traditional crutch of Christian monarchs—and were soon minting coins in collusion with local archbishops.

The story that follows starting in the late 800s is dominated by the figure of Alfred the Great, who united the Anglo-Saxons against the Scandinavian invaders. This was great propaganda. It was a clear rallying cry of us versus them, natives versus invaders, and it seems to have worked. But it also obscured a much more complicated reality of Englishmen picking up Scandinavian habits, and Scandinavians making English culture their own. For example, English words like "Yule" and "law" were never spoken by the Christian monarch Alfred but were rather coined in the more northern Norse-occupied territories around York.

The truth of the matter is that states were consolidating once again, renewing the process that had disintegrated in the generations after Charlemagne. Throughout the 900s, kings pagan and Christian alike found new ways to consolidate power and reap profits from the land. Taxation became more complex, military service increasingly codified. England was torn between Norse and Anglo-Saxon superpowers centered in Wessex and Yorkshire, while the Norse warlords in Francia successfully infiltrated indigenous political structures, eventually consolidating into the duchy of Normandy and later making a successful bid for the English crown in 1066. Scandinavia witnessed similar developments, as Norway, Denmark, and Sweden gradually consolidated into kingdoms, with Norway and Denmark occasionally claiming hegemony over each other.

So by and large I'd say that vikings were keeping up with the Joneses. When the political situation in Western Europe disintegrated, Scandinavians were on hand to exploit the opportunity. And when Western Europeans started getting their act together, Norse-speaking lords taught them a thing or two about urbanization and taxation, which they'd pioneered at places like Dublin, Ireland, and Birka, Sweden.

Later generations—colonials scratching a living in distant Iceland—would remember this as a golden age, and they'd write some amazing and quite frankly enchanting stories about it. Much much later readers would see the seed of democracy in these sagas, but to be honest, much of that was romantic myth. The sagas record a violent, highly stratified society, eking out an impoverished living in which no one was able to claim enough wealth to call himself a king, and whenever anyone threatened to do so, his neighbors persecuted him ruthlessly. I'm not inclined to see these stories as evidence of a Germanic drive toward democracy (although some scholars have certainly read them this way), especially given the cut-throat competition among aspiring aristocrats that is more clearly attested in the contemporary documents from Western Europe.

I hope that answers your question. I'd encourage you to imagine Norse leaders as well as their Western European peers as opportunistic men with an eye toward realpolitik, making pragmatic choices to build or at least to maintain their power in a volatile and uncertain time, rather than using modern (and mostly post-Enlightenment) ideals of justice and democratic government as a lens for understanding their actions.