The story of Jesus casting out the demons from a man and into a herd of swine is well-known, but where it happened, and what it means, are more contentious questions and deserve fuller analysis and discussion.
The longest and most detailed account of Jesus and the Gadarene swine is found in the Gospel of Mark, 5:1-20. This is hardly surprising, since of all three of the Synoptic Gospels (Matthew, Mark, and Luke) which tell this story, Mark is the one most focused on Jesus’ supernatural talents for driving away evil spirits and exorcising demons. Matthew and Luke, who followed Mark’s text, also recount the story, but it is in Mark that we find the most thorough treatment of the event.
5:1 And they came over unto the other side of the sea, into the country of the Gadarenes.
This is how it’s told in Mark; but in the Gospel of Matthew, which was written after Mark, we are told that Jesus crossed the Sea of Galilee to ‘the country of the Gergesenes’. But apparently ‘Gergesenes’ isn’t right either: it’s probably a copyist’s error. But so is ‘Gadarenes’. So where, if not to the country of the Gadarenes and not to the country of the Gergesenes, did Jesus and his disciples go?
The most likely answer is to the country of the Gerasenes: i.e., Gerasa, a Greek town thought to be the same as the similarly-named Kersa on the east shore of Lake Tiberias. This is just five miles from Capernaum, where Jesus had previously been, so it makes sense. However, the Dictionary of the Bible favours Gadara, on the Yarmuk river about six miles south-east of the Lake of Galilee, so this remains a possibility.
5:2 And when he was come out of the ship, immediately there met him out of the tombs a man with an unclean spirit, 5:3 Who had his dwelling among the tombs; and no man could bind him, no, not with chains: 5:4 Because that he had been often bound with fetters and chains, and the chains had been plucked asunder by him, and the fetters broken in pieces: neither could any man tame him.
5:5 And always, night and day, he was in the mountains, and in the tombs, crying, and cutting himself with stones.
Such actions suggest that the ‘unclean spirit’ possessing the man may be the result of what we would now know as mental illness. Of course, it isn’t necessary to seek a non-supernatural explanation for the ‘spirit’ infesting the man, and we might simply take it at face value. 
But it’s worth bearing in mind that being possessed by demons or devils is more likely to be a sign of mental illness than sexual promiscuity, and this is the root of the misunderstanding over the ‘seven devils’ that Jesus cast out of Mary Magdalene. (We discuss the story of Mary Magdalene here.)
5:6 But when he saw Jesus afar off, he ran and worshipped him, 5:7 And cried with a loud voice, and said, What have I to do with thee, Jesus, thou Son of the most high God? I adjure thee by God, that thou torment me not.
5:8 For he said unto him, Come out of the man, thou unclean spirit.
5:9 And he asked him, What is thy name? And he answered, saying, My name is Legion: for we are many.
‘Legion’ has led to some misunderstanding over the centuries. It is often assumed to be the name of the spirit, most probably because it’s capitalised in many translations. But a ‘legion’ was a group of soldiers gathered together in a Roman army. So ‘My name is Legion’ should be analysed as metaphorical rather than a literal statement of name.
5:10 And he besought him much that he would not send them away out of the country.
5:11 Now there was there nigh unto the mountains a great herd of swine feeding.
5:12 And all the devils besought him, saying, Send us into the swine, that we may enter into them.
The words ‘all the devils’ lends credence to the interpretation of ‘Legion’ as denoting the sheer number of demons or spirits inhabiting this one man. It’s going to take a ‘great herd of swine’ or pigs to offload them all from the man.
5:13 And forthwith Jesus gave them leave. And the unclean spirits went out, and entered into the swine: and the herd ran violently down a steep place into the sea, (they were about two thousand;) and were choked in the sea.
5:14 And they that fed the swine fled, and told it in the city, and in the country. And they went out to see what it was that was done.
As so often in the Gospel of Mark, word of Jesus’ powers spreads through reports of his actions, rather than Jesus’ own announcement of his divinity. Here, the swineherds all flee to tell those in the city (whether Gadara or Gerasa) what they have witnessed.
5:15 And they come to Jesus, and see him that was possessed with the devil, and had the legion, sitting, and clothed, and in his right mind: and they were afraid.
Note: ‘had the legion’, i.e., the many unclean spirits inhabiting him. The people who witnessed this are awed by what they have seen, to the point of being frightened that such power exists.
5:16 And they that saw it told them how it befell to him that was possessed with the devil, and also concerning the swine.
5:17 And they began to pray him to depart out of their coasts.
5:18 And when he was come into the ship, he that had been possessed with the devil prayed him that he might be with him.
5:19 Howbeit Jesus suffered him not, but saith unto him, Go home to thy friends, and tell them how great things the Lord hath done for thee, and hath had compassion on thee.
5:20 And he departed, and began to publish in Decapolis how great things Jesus had done for him: and all men did marvel.
You might have thought Jesus would be glad to have a grateful and staunch follower added to his retinue, but once again we find him doing the surprising thing and sending the man home. However, in keeping with so much of the Gospel of Mark – and, to a large extent, the other Synoptic Gospels – we find Jesus’ importance spreading through word of mouth, and Jesus perhaps realised that this man was more valuable to his cause if he stayed home in this obscure town and told everyone of how Jesus had cured him.
There’s also the obvious but significant point that the people of Gerasa/Gadara would know the man Jesus had cured, and would have thought of him as ‘afflicted’ or ‘touched’ or a ‘madman’ (and so on). Seeing him cured, and hearing of how it was accomplished, would more convincingly make the case for Jesus’ divinity among those who had not encountered him for themselves.