Strictly speaking, in
modern usage, a person who refuses to
recant Christian (especially
Catholic)
faith even though he or she is
killed for that
faith by non-believers. More loosely, a martyr can be one who makes a great
sacrifice for a
belief -- perhaps as
mundane as a
business executive losing his or her
job because of a
difference of opinion on
strategy.
Roman martyrs
The word comes from the ancient Greek martus, which simply means "witness," shaded toward the religious sense -- one who has seen literal works of God and wants to tell others about them. The term acquired its modern sense fairly quickly; it came into common use in reference to the Apostles of Jesus Christ, at a time when bearing witness to his apparent miraculous activities was synonymous with bringing Roman persecution on oneself. The first martyr, by this standard, was St. Stephen, a deacon appointed by the Apostles to help care for the poor and aged, who was stoned outside Jerusalem for preaching Christianity.
The tradition of martyrdom grew as the Christian Church grew in prominence under the Roman Empire -- the famous Christians who were thrown to the lions were considered martyrs, but so were those on the fringes of the empire who were killed by Roman authorities with much less ceremony. The term expanded somewhat, especially in the third century, to include people who weren't killed for their Christianity but were stripped of offices and lands, and sentenced to slave labour.
The veneration that came with martyrdom led some people actually to seek it out, which became a problem for the Church. A series of Christian leaders condemned the practice of Christians' surrendering themselves to the authorities for the purpose of being killed and achieving martyrdom -- this was seen as a sort of pridefulness, not true service to God. There was some dissension on the point, though; Lactantius, a Church leader in the fourth century, suggested martyrdom as a solution for lapsed Christians trying to be re-admitted to the faith.
Martyrdom in the Middle Ages and later
After Christianity became the official religion of the Roman Empire, martyrdom became much less common. It arose again, however, after the empire fell; although there were repeated military Crusades to try to take the Holy Land back from Muslim control, many (rather foolhardy) evangelists tried to get it back by going to Jerusalem and arguing religion with the non-Christians who held it.
This didn't generally work very well, and many well-meaning preachers of Christianity were martyred there. Meanwhile, the martyr meme spread into other religions, especially Islam.
A third wave of martyrdoms came with the European expansion into the New World, when an enormous number of Catholic missionaries, especially from Spain and France, were killed by native people for their insistence on proselytizing.
Modern martyrdom
The traditional definition of martyrdom has it that being killed for one's belief should essentially be a passive thing -- one goes about one's business, until one's religious enemies interfere. But as religious conflict grows, seeking out martyrdom is becoming increasingly common. Suicide bombings, travelling to fight in holy wars, and yes, flying airplanes into buildings, are seen as a sort of pro-active martyrdom, guaranteeing one's route into Heaven or Paradise by dying in service to God.
Political leaders can take advantage of this perception by promoting the public veneration of people who do such things (whether they die in the service of a deity or in pursuit of particular principles). Palestinian suicide bombers have their portraits turned into posters, their praises sung in mosques, and their families supported with pensions. American soldiers who die in combat get their portraits on CNN, their funerals reported in the newspapers, and their families supported with pensions.
It's an open question whether all those people really are martyrs; religious scholars of most persuasions seem to think not. We'll find out on Judgment Day (if there is one).