This is an excellent question, Groundhog. I once had a conversation with a guy who told me he'd read that women hated insecurities in men... then proceeded on to tell me how "good-looking" he was, that tons of women wanted him, and that he was a "mack daddy." Ahem.
For me, at least, this is NOT the kind of "confidence" I'm looking for.
I would like someone who is fairly comfortable with who he is and is able to trust other people. For example, if I were in a relationship with someone and I told him about something that had happened at work, such as, a male co-worker saying something that made me laugh, I would want him to be confident enough to be like, "Oh, that really was funny," instead of, "What are you doing talking to some other guy? Are you cheating on me? Are you thinking of leaving me?" etc.
I know it works both ways (women can be like this too) but over time (and unfortunately, experience), I've learned to run from anyone who has this kind of low self-esteem.