Dora Bas Rivka Silver O'H
While David is willing to be punished for his own sins, he asks that G-d not chastise him in anger, but rather a loving (albeit firm) correction, like when a parent punishes a child. David asks for consideration because when he wrote this Psalm he was weak and unable to bear much more affliction. He is already afraid that he will die from his poor health; how long will G-d wait to heal him?
David asks G-d to stop his disease and allow him to be healed, not because David deserves it, but because G-d is merciful. If David dies, he would no longer be able to serve G-d (compare with Psalms 115:17, that the dead do not praise G-d) so he desires to live.
David's moans weaken him and his tears soak his bed. His vision has dimmed, as his persecutors have prematurely aged him. He warns his enemies that they'd better get away, now that G-d has heard his prayers. (Prayers are most effective when accompanied by sincere tears - see Talmud Baba Metzia 59a.) David's enemies will be terrified when they see he has recovered and they will be ashamed when they approach David to beg his forgiveness.
David's note to the conductor indicates that this Psalm is meant to be accompanied by the sheminis, an eight-stringed instrument. This Psalm is recited almost daily as part of the Tachanun prayer.