I hear the term antihero getting thrown around pretty loosely. What people seem to mean when they use the word is a protagonist who -- while still embodying the classically heroic traits of determination, competence and honor -- fights against the standard social structures or status quo rather than fighting to support them. This sort of antihero seems more and more commonplace as people become less and less enchanted with the status quo. It is, of course, easy to root for a hero who fights against the social structures that seem to be holding us back or seems to be making the world a darker and more unkind place. I came across a very enjoyable presentation of this sort of antihero recently in Grant Morrison and J.G. Jones's Marvel Boy, a teenage alien (completely human-looking though his alien DNA has been spliced with that of a cockroach) who, in retaliation for the death of his crew, let's The Man have it but good.
What I hardly ever see is an antihero in the alternate sense. That is, a protagonist who embodies characteristics that are antithetical to those of the classic hero. Little wonder there, as it is difficult to get a readership or an audience to root for a character who is cowardly, vain, greedy or cruel even if that character is also compelling in other ways. This sort of antihero is not essentially equivalent to a villain as, ironically, really good villains tend to share many admirable traits with the hero (who wants to see a great hero go up against a villain who doesn't match up to him or her?). One fine example of an antihero of this variety is Verloc in Conrad's The Secret Agent, who possess relatively few (if any) heroic characteristics, but remains a riveting protagonist.
A truly shocking example of antiheroism (the most shocking I've ever seen, anyway) is in Thomas Harris's Red Dragon. Will Graham is, on the surface, a classic flawed hero, dragged back into a heroic role he left behind because it requires him to use an ability which is far more a curse than a gift. For practically the whole book we follow Will's pursuit of a serial killer and are in his corner the entire time. At the very end of the book, however, during the final confrontation, Will does something I've never seen another hero (or antihero) do. I'm not going to say what it is (that would ruin Mr. Harris's powerful climax), but I will say that it can only be seen in the book as neither movie version (Manhunter nor Red Dragon) would dare to reproduce it (neither, for that matter, does the Wikipedia summary I linked to above). Only in a truly heroic milieu, it seems, can you really plumb the depths of antiheroism.