Thomism is a Medieval school of philosophy that arose specifically as a legacy of the work and thought of the 13th Century philosopher and theologian St. Thomas Aquinas. His "Summa Theologica" is often considered second only to the Bible in importance to the Roman Catholic Church, and arguably one of the most influential philosophies of all time.
Aquinas worked to create a philosophical system which integrated Christian doctrine with elements taken from Aristotelianism, augmenting the Neo-Platonic view of philosophy (which, after St. Augustine, had become tremendously influential among medieval philosophers), with insights drawn from Aristotle. He was instrumental in moving the focus of Scholastic philosophy away from Plato and towards Aristotle.
He was greatly influenced by his reading of earlier and contemporaneous Islamic philosophers, especially the works of Avicenna, Al-Ghazali (1058 - 1111), and Averroes (although he explicitly rejected Averroes' primary conclusions and themes). He also drew on the works of the prominent medieval Jewish philosphers Avicebron (1021 - 1058) and Maimonides, and in turn he influenced later Jewish philosophy.
Aquinas taught that both faith and reason discover truth (conflict between them being impossible since they both originate in God), and that reason can, in principle, lead the mind to God. He offered five proofs for the existence of God, including the Cosmological Argument (based on Aristotle's concept of the "unmoved mover") and the Teleological Argument (which is similar to the modern idea of "intelligent design"). See the secion on Philosophy of Religion for more discussion of these.
The Thomistic School is distinguished from other schools of theology chiefly by its doctrines on the difficult questions relating to God's action on the free will of Man, God's foreknowledge, ther nature of grace (he held that grace was not due to Man's nature, but was granted to Man by God from the beginning), and predestination (the idea that God has appointed and pre-ordained from eternity all events occurring in time).
The Dominican religious order, of which Aquinas was a member, quickly adopted his ideas as an official philosophy of the order, and the Dominicans always remained his most ardent supporters, through to the 16th Century. The Franciscan order, on the other hand, including John Duns Scotus, Henry of Ghent (c. 1217 - 1293) and Giles of Rome (c. 1243 - 1316), vehemently opposed Thomism. Some of his theses were condemned in 1277 by the important ecclesiastical authorities of Paris and Oxford, although this condemnation was revoked after Aquinas was canonized in 1323. William of Ockham and his adherents also expressed strong opposition to Thomism, as did the later Jesuit Molinists (named after the movement's founder, the 16th Century Jesuit theologian Luis de Molina), notably Robert Bellarmine (1542 - 1621), Francisco Suárez (1548 -1617) and Francisco de Lugo (1580 - 1652).
In the late 19th Century, Pope Leo XIII (1810 - 1903) attempted a revival of Thomism (Neo-Thomism), emphasizing the ethical parts of Thomism, and this held sway as the dominant philosophy of the Roman Catholic Church until the Second Vatican Council in 1962, and remains a vibrant and challenging school of philosophy even today.
Analytical Thomism is a recent minor philosophical movement which promotes the interchange of ideas between Thomism and modern Analytic Philosophy.