Total depravity defined
The Canons of Dort define total depravity pretty well in section 3:
All people are conceived in sin and are born children of wrath, unfit for any saving good, inclined to evil, dead in their sins, and slaves to sin. Without the grace of the regenerating Holy Spirit they are neither willing nor able to return to God, to reform their distorted nature, or even to dispose themselves to such reform. ... To hunger and thirst for deliverance from misery and for life, and to offer God the sacrifice of a broken spirit is characteristic only of the regenerate and of those called blessed.
Some think total depravity means that man is as evil as he possibly could be or that there is no sense in which man could ever do anything considered good. The Heidelberg Catechism may seem, at first glance, to support this notion in question 8:
Q. But are we so corrupt that we are totally unable to do any good and inclined toward all evil?
A. Yes, unless we are born again by the Spirit of God.
But that's not the whole picture. Consider what Louis Berkhof says in his systematic theology:
The heathen religions testify to a universal consciousness of sin and of the need of reconciliation with a Supreme Being. There is a general feeling that the gods are offended and must be propitiated in some way. There is a universal voice of conscience, testifying to the fact that man falls short of the ideal and stands condemned in the sight of some higher Power.
...
[Total depravity] does not imply: (1) that every man is as thoroughly depraved as he can possibly become; (2) that the sinner has no innate knowledge of the will of God, nor a conscience that discriminates between good and evil; (3) that sinful man does not often admire virtuous character and actions in others, or is incapable of disinterested affections and actions in his relations with his fellow-men; nor (4) that every unregenerate man will, in virtue of his inherent sinfulness, indulge in every form of sin.
This is confirmed in Dort, but note well the final sentence:
There is, to be sure, a certain light of nature remaining in all people after the fall, by virtue of which they retain some notions about God, natural things, and the difference between what is moral and immoral, and demonstrate a certain eagerness for virtue and for good outward behavior. But this light of nature is far from enabling humans to come to a saving knowledge of God and conversion to him.
The Westminster Confession (16.7) has a balanced discussion of the "goodness" of unregenerate man's works in light of the doctrine of total depravity:
Works done by unregenerate men, although for the matter of them they may be things which God commands; and of good use both to themselves and others: yet, because they proceed not from an heart purified by faith; nor are done in a right manner, according to the Word; nor to a right end, the glory of God, they are therefore sinful and cannot please God, or make a man meet to receive grace from God: and yet, their neglect of them is more sinful and displeasing unto God.
From these statements in the reformed confessions, we can see three aspects of what total depravity entails. They are intertwined and closely related, but I think keeping them separate in one's mind is helpful for understanding what the doctrine is about:
Man has rebelled completely against God. In this state of rebellion, he is enslaved to sin and Satan from the moment of his first breath, and he deserves condemnation.
Man is totally unable to turn away from this state of rebellion by an act of the will. He needs God's enabling grace in order to exercise faith.
It is only when man has faith that he is able to do "good works."
The Biblical witness
All quotes are NET.
Total rebellion against God
But the Lord saw that the wickedness of humankind had become great on the earth. Every inclination of the thoughts of their minds was only evil all the time.
Genesis 6:5 (cf. Genesis 8:21)
If you then, although you are evil, know how to give good gifts to your children, how much more will your Father in heaven give good gifts to those who ask him!
Matthew 7:11
Look, I was guilty of sin from birth, a sinner the moment my mother conceived me.
Psalm 51:5
All of us had wandered off like sheep; each of us had strayed off on his own path.
Isaiah 53:6
“There is no one righteous, not even one, there is no one who understands, there is no one who seeks God. All have turned away, together they have become worthless; there is no one who shows kindness, not even one.” “Their throats are open graves, they deceive with their tongues, the poison of asps is under their lips.” “Their mouths are full of cursing and bitterness.” “Their feet are swift to shed blood, ruin and misery are in their paths, and the way of peace they have not known.” “There is no fear of God before their eyes.”
Romans 3:10-18 (a slew of Old Testament quotations)
Sin entered the world through one man and death through sin, and so death spread to all people because all sinned.
Romans 5:12
And although you were dead in your transgressions and sins, in which you formerly lived according to this world’s present path, according to the ruler of the kingdom of the air, the ruler of the spirit that is now energizing the sons of disobedience, among whom all of us also formerly lived out our lives in the cravings of our flesh, indulging the desires of the flesh and the mind, and were by nature children of wrath even as the rest…
But God, being rich in mercy, because of his great love with which he loved us, even though we were dead in transgressions, made us alive together with Christ—by grace you are saved!
Ephesians 2:1-5 (cf. Colossians 2:13)
All is pure to those who are pure. But to those who are corrupt and unbelieving, nothing is pure, but both their minds and consciences are corrupted.
Titus 1:15
Total inability apart from God's grace
But there is little hope for you ever doing good, you who are so accustomed to doing evil. Can an Ethiopian change the color of his skin? Can a leopard remove its spots?
Jeremiah 13:23 (cf. Jeremiah 17:9)
I will give you a new heart, and I will put a new spirit within you. I will remove the heart of stone from your body and give you a heart of flesh.
Ezekiel 36:26
Jesus replied, “I tell you the solemn truth, unless a person is born from above, he cannot see the kingdom of God.” Nicodemus said to him, “How can a man be born when he is old? He cannot enter his mother’s womb and be born a second time, can he?” Jesus answered, “I tell you the solemn truth, unless a person is born of water and spirit, he cannot enter the kingdom of God.”
John 3:3-5
No one can come to me unless the Father who sent me draws him, and I will raise him up at the last day.
John 6:44
Why don’t you understand what I am saying? It is because you cannot accept my teaching. 44 You people are from your father the devil, and you want to do what your father desires. He was a murderer from the beginning, and does not uphold the truth, because there is no truth in him. Whenever he lies, he speaks according to his own nature, because he is a liar and the father of lies. ... The one who belongs to God listens and responds to God’s words. You don’t listen and respond, because you don’t belong to God.
John 8:43-44,47
The unbeliever does not receive the things of the Spirit of God, for they are foolishness to him. And he cannot understand them, because they are spiritually discerned.
1 Corinthians 2:14
The inadequacy of the light of nature
For since the creation of the world his invisible attributes—his eternal power and divine nature—have been clearly seen, because they are understood through what has been made. So people are without excuse. For although they knew God, they did not glorify him as God or give him thanks, but they became futile in their thoughts and their senseless hearts were darkened.
Romans 1:20-21
The inadequacy of the law
For God achieved what the law could not do because it was weakened through the flesh. By sending his own Son in the likeness of sinful flesh and concerning sin, he condemned sin in the flesh, 4 so that the righteous requirement of the law may be fulfilled in us, who do not walk according to the flesh but according to the Spirit.
Romans 8:3-4
He saved us not by works of righteousness that we have done but on the basis of his mercy, through the washing of the new birth and the renewing of the Holy Spirit
Titus 3:5
The necessity of faith in God to do good works
I am the vine; you are the branches. The one who remains in me—and I in him—bears much fruit, because apart from me you can accomplish nothing.
John 15:5
Whatever is not from faith is sin.
Romans 14:23
Those who are in the flesh cannot please God.
Romans 8:8
Now without faith it is impossible to please him, for the one who approaches God must believe that he exists and that he rewards those who seek him.
Hebrews 11:6