The Romans had been lusting after Britain for quite some time. Julius Caesar had made two passes according to Dio Cassius in 55 and 54 BCE, but did not make much headway. The first only established a beachhead, but the second established a king friendly to Rome. However, Caesar had bigger fish to fry and headed back to Rome and glory leaving Britain alone in the mists.
Caligula had tried again in 40 CE, and had a lighthouse built in preparation at Gesoriacum, modern Boulogne-sur-Mer. However, according to the tale told by Suetonius in The Twelve Caesars, he made less progress than Caesar. According to this story, he had the legions line up in battle formation on the Gallic shore facing Britain against the English Channel. He then declared war on Neptune, the god of the sea, and commanded his soldiers to attack the waves. After they had done their duty and stabbed the errant waves, he had the drenched and probably bewildered men collect shells, which were shipped to the capital as spoils of war. There is some doubt cast upon the veracity of this story, but it goes to show Caligula’s reputation for unstable behavior, can be considered as possible.
It was Claudius who finally added Britain to the Roman Empire in 43 CE. The invasion was ostensibly to restore Rome’s ally Verica, the exiled King of the Atrebates. However, once Rome landed and began to meddle with local politics it was a short step until they took over. They sent 20,000 men under the command of Aulus Plautius. The force included the future emperor Vespasian, who commanded one of the legions.
In short order, they had the resistance beaten and Claudius himself came to receive the official surrender. Suetonius reports Claudius received the surrender of the British kings with no blood shed, and the Claudian arch shows him receiving the surrender of eleven kings. Claudius declared Camulodunum, modern Colchester, as the new capital then returned to Rome.
He left his legions who continued to mop up any resistance in his absence. However, only time would tell if Britain was truly subdued. For more on this please see this post: http://www.historynaked.com/boudiccas-revolt/