A bill of attainder is an act of legislature declaring a person or group of persons guilty of some crime, and punishing them, without benefit of a trial. The United States Constitution forbids Congress to pass any bills of attainder.
The word "attainder", meaning "taintedness", is part of English common law. Under English law, a criminal condemned for some crime could be declared "attainted", meaning that his entire family was "tainted" with his crime. Once declared attainted, the criminal's entire family would be barred from inheriting the property of the criminal, which would consequently revert to the Crown. Any peerage titles would also revert to the Crown. The convicted person might also be punished in other ways; for example, in the case of attainder for treason, he could be executed.
Bills of attainder evolved into a convenient way for the King to convict subjects of crimes, and confiscate their property, without the bother of a trial--and without the need for a conviction, or indeed any evidence at all.
In some cases (at least regarding the peerage) the Crown would eventually re-grant the convicted peer's lands and titles to his heir. It was also possible, as political fortunes turned, for a bill of attainder to be reversed. This might even happen long after the convicted person was dead.
Bills of attainder were used through the 18th century in England, and were applied to English colonies as well. One of the motivations for the American revolution was anger at the injustice of attainder--though the Americans themselves used bills of attainder to confiscate the property of English loyalists (called tories) during the revolution. American dissatisfaction with attainder laws motivated their prohibition in the Constitution.
Within the U.S. Constitution, the clause forbidding attainder laws served two purposes. First, it reinforced the separation of powers, by forbidding the legislature to perform judicial functions--since the outcome of any such acts of legislature would of necessity take the form of a bill of attainder. Second, it embodied the concept of due process, which was later reinforced by the fourth amendment to the Constitution. Any forfeiture of life, liberty or property without due process of law is by definition a bill of attainder.