'Semeia' (singular 'semeion'), which is Greek for 'signs', is the word used to describe Jesus' miracles in the Gospel according to St. John. This word is used as opposed to 'dunameis' ('acts of power' or 'mighty works') in the synoptic gospels, and as opposed to Jesus' preferred term, 'erga' ('work'). A similiar term, 'oth' (a Hebrew term basically meaning a symbolic act) is used in the Old Testament; when the Old Testament is translated into Greek, 'semeion' is used.
So why did John prefer this term? Firstly, there were a lot of miracle workers doing the rounds. It really was like that scene in The Life of Brian, where all the preachers are lined up on their soapboxes. By referring to Jesus' miracles (for want of a better term) as signs rather than miracles, John was elevating Jesus above these people. Secondly, John wanted the miracles to point beyond themselves. He wanted readers to appreciate the meaning and significance of what had happened, rather than just going, "Hey, cool, he raised Lazarus from the dead," and leaving it at that. Rather than just being amazing acts in themselves, John wanted the semeia to be symbolic of greater truths. They have been described as 'parables in action'; performing a similar function to Jesus' teachings, but in a much more attention-catching way. John's gospel is a lot less straightforward than the synoptics - which is not to say that the synoptic gospels are all history and no theology; that would be blatantly untrue - but in the same way that John's prologue goes above and beyond the birth narratives of Matthew 1 and Luke 1, John's semeia go above and beyond the synoptic miracles.
The primary purpose of the semeia is the same as the purpose of the whole gospel: "This was written so that you may believe that Jesus is the Christ, the Son of God; and that believing you may have life in Him" (John 20:31). This is reinforced by the fact that most semeia end with some onlookers developing faith, and some remaining unbelievers. However, there are other purposes: symbolism into the events of Jesus' life (particularly his crucifixion), his person and the Kingdom of God - referred to in the fourth gospel as "eternal life" - have been read into the semeia. For instance, there are alleged parallels between the raising of Lazarus and Jesus' own resurrection, while his comment while turning water into wine about "my time has not yet come" is generally considered to refer to his crucifixion. Signs are often linked to and illustrate the I am sayings; this casts further light on Jesus' person and work. For instance, he says "I am the resurrection and the life" while during the raising of Lazarus, "I am the bread of life" during the feeding of the 5,000 and "I am the light of the world" while healing the man born blind. The theme of Jesus bringing life - specifically eternal life - is also clear, particularly during the raising of Lazarus and the feeding of the 5,000, and particularly for those scholars who see Jesus' resurrection as a semeion.
There is some controversy as to which events constitute semeia. Many commentators want there to be seven signs, as seven is the perfect number in theology, and to link with the seven 'I am' sayings. It would also make sense because the number seven crops up in the Apocalypse of St. John. However, there may be more or fewer. It is generally agreed that the following are semeia:
Some scholars argue that the following should be classed as semeia: | 
