Total Depravity is the erroneous teaching that man is born so totally depraved that he cannot have a desire to seek out God without the intervention of God. What this means is that we cannot chose to believe in God, but that God must first draw us by his Spirit in order to do so. However, there is not anywhere in the Bible that teaches this. Though there is a verse that says we cannot come to Christ unless God draws us (John 6:44), the teaching there is that if we are truly desiring to follow God that he will lead us to Christ. It is important that one understand that total depravity is a doctrine that goes way beyond the biblical teaching that we have a sinful nature. It doesn’t just imply that we are born with a sinful nature, which is true, but that we are totally depraved in all aspects of our being as well.
Total Depravity is also one of the five points of Calvinism. The Calvinist position is that God must first regenerate our nature so that we will then come to him. In other words, we cannot pick God, but must hope we are one of the lucky ones who God decides to choose. Though the definition of total depravity sometimes varies among Calvinists, this is the basic premise held by most of them. Many also believe that we cannot do anything good at all in our fallen state. They insist that this is because our nature is so depraved that we are incapable of any genuine good without God first changing our nature. The problem with this teaching is that it is contradictory to much of the Bible and paints an ugly, distorted picture of the loving God we worship.
For more information on this false teaching, read Total Depravity and Free Will by Robert Alan King.