Lady Macbeth is able to persuade Macbeth to kill Duncan because she appeals not only to his ambitious nature but attacks his manliness. She asks him, "...Wouldst thou have that/Which thou esteem'st the ornament of life,/And live a coward in thine own esteem;..." She questions his love for her and hers for him if he can't go through with this plan. Lady Macbeth then asks why he bothered to tell her about the plan at all if his ambition wasn't greater than his cowardice. She offers to help him to kill Duncan, suggesting she as a woman is stronger than Macbeth. He's unable to take these insults and agrees to kill Duncan. Her last appeal asks him,

What cannot you and I perform upon
The unguarded Duncan? what not put upon
His spongy officers; who shall bear the guilt
Of our great quell?

The Macbeths are very close, and Lady Macbeth uses it to tell Macbeth that there's nothing the two of them can't accomplish together. She hits Macbeth with everything she can possibly think of, and it works. Macbeth's doubts are no match for her accusations.

Macbeth is persuaded for a few reasons.  First of all, he is tempted by his own ambitions to become king so her persuasion is easily effective.  She's convincing him to do something that will make him king which is what he wants.  Also, Lady Macbeth has the emotional power of a wife over her husband.  When she first greets him in Act I Scene 5, she flatters him with titles, including hinting at the future title of king.  Then, in Act I Scene 7, she also baits him by taunting him and questioning his manhood, calling him a coward, etc. if he doesn't follow through with the plan to kill King Duncan.  Macbeth gives in to all of these forms of persuasion and goes along with his wife's plan.