To first appearances, James Norrington is a typical stuffy English naval officer of his age, who values the rule of law above everything - and HATES pirates. However, there's much more to him than that.
He's a man who can stand back and let the woman he loves go to someone else, even though she's already promised to marry him, because he knows she doesn't love him and would never be happy with him, and can even give his blessing of sorts to the man she's chosen. He's the most gracious loser in a romantic triangle ever.
He's also a man who is capable of coming to respect and perhaps even like one of the pirates he's sworn to stamp out, and of bending the rules to allow that pirate to escape - giving Jack and the Pearl a day's head start, though he's fully aware of the Pearl's reputation as the fastest pirate ship in the Caribbean.
He's the first to draw his sword when battle starts, and doesn't stop fighting until the fighting's done. He does what he believes to be right at all times, even when doing so costs him something he wants dearly, or involves not quite following the letter of the law, though he always upholds its spirit.
Simply put, if the movie had told another story, James Norrington would have been the hero.