Prevenient grace refers to the grace of God in a person's life that precedes conversion (or salvation). The word "prevenient," considered an archaic term today, was common in the King James english and simply means to "go before" or "precede." Likewise, it is sometimes called "preventing" grace (from prevenient) with the same meaning.
In Reformed Theology, it is the particular grace which precedes human decision -- a salvific grace prior to, and without reference to, anything we have done. See Irresistible grace, sometimes called efficacious grace or the effectual call.
In Arminianism and Wesleyanism, it is a grace that offsets the noetic effects of the Fall, restores man's free will, and thus enables every person to choose to come to Christ or not. There are two forms of this view:
In Romanism (i.e., Roman Catholic), it is an assisting grace which aids people who choose to co-operate in justifying themselves. See Council of Trent (Sess. VI, cap. v).