Who were the Vikings?
First of all, the word Viking is probably more something you do that something you are. To go viking was to go raiding or travelling. The Scandinavians who caused Europe so much trouble were also not just one group - and indeed they spent plenty of time beating each other up, as well as the English, Irish and French.
This handy map shows that the Scandinavians went different places, though it doesn't show the full story. Actually, the Swedes mainly concentrated on Russia and the east. The Norwegians mainly invaded Ireland - and it was the Danes that caused England all the trouble. The Danes and Norwegians also ravaged northern France, including beseiging Paris. In 911, a viking called Rollo was granted Normandy by the French King, in a last desperate attempt to find an answer to the terrible raids and destruction.
Why did they come?
The Danes may have gone raiding for a number of reasons - though we don't really know. It could be because they wanted land, and were suffering from overpopulation; they may have been looking for revenge for Charlemagne's bloody attempts to impose Christianity on them; or maybe they just found raiding an easier way to make a buck than farming. Or maybe the death of the strong Danish King Horik I in 854 removed the only central authority that could prevent the departures. What ever the reason, they kept coming.
The first Danish raids
The first recorded raid was 789, entered in the Anglo saxon Chronicle:
. . . came the first three ships of the Northmen from Horthaland. The reeve rode there, and meant to force them to the King's dwelling because he did not know what they were; and then he was killed.
Then in 793 they ravaged Lindisfarne, and raids continued in a sporadic manner until the 840's, gradually increasing in severity. The Kings of Wessex Aethelwulf and Aethelbald fought them, and sometimes won. But the raids, though devastating, did not basically stop the normal operation of the kingdom. And then in 855 the Great Heathen Army came to stay and everything changed.
The Great Heathen Army
In 866 the sons of Ragnar Lothbrok took York and Northumbria. In 869, they did the same in East Anglia, and martyred it's king, Edmund. In 870, it was quite clear that it was time for Wessex and Mercia to fall.
The Anglo Saxon Chronicle
So much of what we know about these times are through the Anglo Saxon Chronicle that I thought I'd put up a table of the bits of the entries concerning Vikings, for the period covered by Episode 6 in the History of England podcast. You can easily see the whole thing on t'internet if you are too mean to buy it - try the link below, for example.