SAP is also short for Sveriges socialdemokratiska Arbetarparti or in english, The Swedish Social Democratic Party. It was formed in 1889 by workers who were tired of the injustice they suffered in society. On the party homepage they claim to have about 160.000 members, which would make them the largest political party in Sweden.
SAP got their first member in parliament in 1896. His name was Hjalmar Branting, for a long time an important person in Swedish politics. At that time, not everybody had the chance to vote, so the issue of public rights to vote were one of the first things the party worked for. In 1902, after a settlement with the conservatives, all men were allowed to vote from the age of 24. Women were still not allowed to vote, their right came after World War I.
In 1932 SAP won the election and stayed in power for the next 44 years. The party have had many chairmen that have made themselves known internationally, for instance Olof Palme who was assasinated in the streets of Stockholm in 1986. The current chairman is Goran Persson, who is also the Prime Minister of Sweden.
Like many other labour parties in Europe, SAP has over the years softened their socialist demands that they worked so hard for in their early days. Now they seem mainly to strive for status quo in society. To keep both workers and capitalists happy in the same time, that is.
In Sweden, a person who sympathizes with SAP would be called a "sosse". It's sometimes used as a slightly disparaging term. For instance, a "gråsosse" (or in english, "gray sosse") is a person who is a bit of a bureaucrat and drive ugly cars.