The three main rivals for the throne were:
When Edward the Confessor died, Harold Godwinson, Earl of Wessex, was immediately crowned king and became Harold II. The royal council, known as the Witan, supported him. He gathered an army to defend the kingdom.
Edgar Ethling was Edward’s closest direct relative, but this was a distant connection as he was Edward’s father’s grandson by his first marriage. Edgar was weak and lacked credibility due to being a child in exile in Hungary.
Harald Hardrada was king of Norway. He invaded Yorkshire with a fleet of ships, but was defeated and killed by Harold's army at the Battle of Stamford Bridge.
While Harold II was in the north of England fighting Hardrada, William, Duke of Normandy invaded Sussex. Harold rushed back south to fight him.
On 14 October 1066, Harold II fought William's army at the Battle of Hastings and lost. Harold was killed, perhaps with an arrow in his eye - although this is disputed by some historians.
William was crowned king of England on Christmas Day 1066, but it took years more fighting to conquer the whole country. His cruellest campaign was the 'Harrying of the North' in 1069, where he slaughtered the inhabitants of the north-east and destroyed their food stores so that even the survivors starved to death.
The Norman Conquest changed the face of England forever. William ruled as unquestioned conqueror and the Saxons became merely an unpaid workforce for their new lords (see The feudal system and the Domesday Book).
The Norman Conquest also changed the history of Europe – adding the wealth of England to the military might of Normandy made the joint-kingdom a European super-power.
In warfare, it was the start of the age of the knight-on-horseback.
You may wish to compare the Norman Conquest with the Glorious Revolution, covered in the UK government through time, the only other successful invasion in England's history after AD1000 – they were very different.