Primarily an Irish custom though not uncommon in other Gaelic nations. The Wake is considered an integral part of the grieving process after the death of a loved one. Family, friends neighbours and the community in general, gather in the house of the deceased to 'Wake' the body.
The term 'Wake' is widely believed to come from the time when stout was served in lead lined cruscáns. It sometimes occured that a person would suffer severe lead poisoning after drinking from these. The body would, temporarily, go into a death like state, after a certain amount of time the body recovers. It was traditional, therefore, that if the cause of death could not be explained, the family would place the body in the house and hope for the person to Wake up!!(Obviously at the time there was no understanding of what was causing the body to display these death-like symptoms)
The body is usually placed in a coffin (an open casket if possible) in a room in the house, the body is not to be left alone from this point until it is removed to the church for burial. It was the job of the women in the community to prepare the body to be "laid out".
Once the body is prepared properly it is time for the keening to start, this consists of the women present crying and wailing. Legend has it that this keening will keep the spirits away until the body could be removed. At some point in the proceedings the rosary will be said.
At the wake there will nearly always be large amounts of food and particularly drink and after a while there will be Irish dancing and games - in celebration of the life of the deceased. The Church has, in the past, made moves to discourage this behaviour, to no avail.
The Modern Wake is very similar to the traditional wake but differs in certain areas. Firstly, an Undertaker will prepare the body, not the communities women, secondly it is very rare to hear the traditional keening at a modern wake - although it does occur. The wake tends to be a far less religiously oriented occasion, it is more about giving the family and friends time to say goodbye. However, as with the traditional wake, the rosary is usually said at some stage.
There is, of course, still plenty of drink and food but the traditional dancing rarely occurs. The whole affair tends to be less structured, with people just tending to chat and drink until early morning.
During the Irish Diaspora (around the time of the Great Famine) as millions left the country, the tradition of the American Wake started. On the evening before a person was to depart (usually to America) a wake like party would be held to bid them farewell. In the majority of cases of emigration, the departee would never be seen again by his family and friends, hence the link to the Wake.
22nd July '04 skow has informed me that wakes are in fact not strictly a Gaelic tradition, they are also performed in Germany and The Netherlands.