A quick overview of the happenings in 1066, to start with:
In January 1066 the English King, Edward the Confessor, died, leaving no heir. As a result, there were three main figures who laid claim to the throne of England - Harold Godwinson, Harald Hardrada, and William of Normandy.
Harold and Edward were brothers-in-law, as the King had married Harold's sister. Harold's position as advisor to Edward (and thus the second most powerful person in England), as well as his relationship with Edward, made him the most obvious successor to the throne. He was crowned King on the day of Edwards burial.
Harald Hardrada was the King of Norway. Hardrada had ruled Norway jointly with his nephew Mangus until 1047 when Mangus conveniently died. Earlier (1042), Mangus had cut a deal with Harthacut, the Danish ruler of England. Since neither ruler had a male heir, both promised their kingdom to the other in the event of his death. Harthacut died but Mangus was unable to follow up on his claim to the English throne because he was too busy battling for the rule of Denmark, and so Edward became the Anglo-Saxon ruler of England. Now with Mangus and Edward dead, Hardrada asserted that he, as Mangus' heir, was the rightful ruler of England. When he heard of Harold's coronation, Hardrada immediately prepared to invade England and crush the upstart.
Across the English Channel, William, Duke of Normandy, also laid claim to the English throne. William justified his claim through his blood relationship with Edward (they were distant cousins) and by stating that some years earlier, Edward had designated him as his successor. To compound the issue, William asserted that the message in which Edward anointed him as the next King of England had been carried to him in 1064 by none other than Harold himself. In addition, (according to William) Harold had sworn on the relics of a martyred saint that he would support William's right to the throne. From William's perspective, when Harold donned the Crown he not only defied the wishes of Edward but had violated a sacred oath. He immediately prepared to invade England and destroy Harold. Harold's violation of his sacred oath enabled William to secure the support of the Pope who promptly excommunicated Harold, consigning him and his supporters to an eternity in Hell.
Hardrada was the first to invade. In mid September, Hardrada's invasion force landed on the Northern English coast, sacked a few coastal villages and headed towards the city of York. Hardrada was joined in his effort by Tostig, King Harold's renegade brother. The Viking army overwhelmed an English force blocking the York road and captured the city. In London, news of the invasion sent King Harold hurriedly north at the head of his army picking up reinforcements along the way. The speed of Harold's forced march allowed him to surprise Hardrada's army on September 25th, as it camped at Stamford Bridge outside York. A fierce battle followed. Hand to hand combat ebbed and flowed across the bridge. Finally the Norsemen's line broke and the real slaughter began. Hardrada fell and then the King's brother, Tostig. What remained of the Viking army fled to their ships. So devastating was the Viking defeat that only 24 of the invasion force's original 240 ships made the trip back home. Resting after his victory, Harold received word of William's landing near Hastings (he landed in a place called Pevensey, where he built his first conquest Castle). And so Harold marched his army south as quick as he could, and eventually lined them up on a hill some five miles from Hastings.
The rest, as they say, is History. The Norman army defeated the English, after Harold fell to an arrow in his eye. William was eventually crowned King.
Interestingly enough, the invasion of 1066 marks the last time that England was invaded.
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Just to correct your question; it was
William who invaded. Harold was the King of England at that time, and it was he who was killed by the arrow.
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EDIT: Just realised I didn't answer.

I'll answer in the morning when I'm more awake. There's a lot of things the Normans did that wouldn't have happened without them (they introduced a new law system, for example, and the main language spoken in court - French), and I can't really think too well on a tired mind.