Norway has a rich and vibrant history, and it dates back to an incredibly long time ago. Unlike the US, Europe has been populated for thousands and thousands of years, and has had many many large civilizations come and go. North America was settled by the first humans perhaps 15,000 years ago, but no significant civilizations were present, unlike Europe.
Norway was first settled in 10,000 BC or so, approximately 12,000 years ago. The inhabitants were supported by fishing and hunting reindeer, but around 5,000-1,500 BC, agricultural settlements began to pop up around southern Norway. The Viking Age began in 793 AD, and the Vikings spread out to Greenland, North American in Newfoundland, Iceland, Scotland, Ukraine, Ireland, Russia and several other places. It was during this time that North America was settled by Europeans… almost 500 years before Christopher Columbus set foot in the Western Hemisphere! There is, however, no evidence of lasting settlements in North America by the Vikings. In 800 AD, the Vikings founded the first cities in Ireland, including Dublin. The Vikings were eventually driven out by the Celts.
During this time period, the Scandinavians spoke Old Norse, and Norse Runes were used in writing. This was the prototype language that is now used in Norway, as well as the other Scandinavian countries. Icelandic, Faroese, Norwegian, Danish and Swedish all came from Old Norse, and have a high degree of mutual intelligibility, especially Norwegian, Danish, and Swedish.
Harald Fairhair is considered to be the first King of Norway. He reigned from 872 to 930 AD, and this is considered to be the first time Norway was a united Kingdom. However, the unification did not last long after his death, and Norway was then ruled in whole or part by various earls in Norway, or by Danish rulers. Beginning in 1015 AD, Christianization happened, abolishing Norse Paganism. After a few years, however, Danish rule had become unpopular enough that Norway was once again unified. in 1130 AD civil war broke out due to unclear succession laws, however, peace was re-established in 1217 after clear succession laws were put in place.
The 1300’s were Norway’s Golden Age. Peace abounded, and trade with other countries flourished. The population of Norway had increased significantly since the 1000’s. In 1349 however, the Black Death had spread to Norway, and killed about a third of the population. Between the Black Death and other plagues, the population decreased to half the starting point by 1400. In 1380, Olaf Haakonsson both the Norwegian and Danish thrones, and in 1397, the Kalmar Union was formed, which combined Denmark, Norway, and Sweden (which at the time also included Finland) into one State. There were several short interruptions, but the Union lasted until 1523. However, after the breakup of the Kalmar Union, Norway and Denmark remained as a single State until 1814. During this time, a series of wars were waged between Denmark/Norway and Sweden.
Denmark/Norway entered the Napoleonic Wars in 1807, with Denmark/Norway allying with France, and Sweden allying with the United Kingdom. The Napoleonic forces eventually lost, therefore Denmark/Norway also lost, and Denmark agreed to cede Norway to Sweden in the course of the war ending. Thus Norway/Sweden began a Union in 1814. The Constitution of Norway was rewritten at this point, and it was signed into law on 17 May, 1814, which is now celebrated as Syttende Mai, Norway’s Constitution Day, akin to America’s Fourth of July. Norway’s Constitution is thus the second oldest constitution currently in use in the world today.
Norwegian support of the Norway/Sweden Union began to diminish in the end of the 1890’s, and in 1905, Norway’s Parliament voted to create a Norwegian consular service, which the King rejected. Thus on 7 June 1905, Parliament unanimously voted to dissolve the Union. A referendum was issued to the people of Norway, with near 100% approval, only 184 people voted against it. (At the time, only Norwegian men had voting rights. Women’s suffrage was introduced in 1913, making it the second country in the world where women had voting rights.) After this dissolution, the Norwegian crown was offered to Denmark’s Prince Carl, whose family controls the Norwegian throne to this day, and Norway became a fully independent country.
WWII began in 1939, and Norway remained neutral in the beginning. On 9 April, 1940, however, German invaded Norway, and occupied it until the end of the war. The rightful government of Norway, including the royal family, escaped to London, where they ruled in exile, until the end of the war. Germany eventually surrendered in 1945, and control of Norway was returned to the Norwegians.
Between 1966 and 1969, it was found that Norway had very large offshore oil reserves. This brought Norway’s economy great fortune, and in 1994, Norway voted against joining the EU, in part due to the fact that Norway was very well off by itself given the massive oil reserves, but also due to the fact that the EU’s fishery laws would negatively affect Norway.
There are many details about Norway’s history that have been left out here, but see the sources below for further reading about these topics. 
Sources:
http://en.wikipedia.org/wiki/History_of_Norway
http://en.wikipedia.org/wiki/Dissolution_of_the_union_between_Norway_and_Sweden
http://en.wikipedia.org/wiki/Treaty_of_Kiel
http://en.wikipedia.org/wiki/Kalmar_Union
http://en.wikipedia.org/wiki/Harald_Fairhair
http://en.wikipedia.org/wiki/Unification_of_Norway
http://en.wikipedia.org/wiki/Dublin
http://en.wikipedia.org/wiki/Viking_Age
http://en.wikipedia.org/wiki/Old_Norse