I have the same sort of casual friendships that most people have, I think; except, of course, being an autistic introvert, I didn't get to experience them until the sixth grade, and then not again until college. In those friendships, it's probably about the same thing as a sexual person having friendships; and can get awkward for the same reasons that it might with a sexual person--you want a casual friendship and they want romance, then you have to break it to them that you're not attracted.
There are closer friendships, and that is something I've only experienced once. I guess they're what you'd be calling "best friends"; but it's not quite like the "best friends" you see in high school or whatever; it's more like... I dunno, I don't think English has a word for it. Oddly enough, Japanese seems to--
http://tvtropes.org/pmwiki/pmwiki.php/Main/Nakama
That sort of thing is probably the closest I can get to "romance"; only there's nothing very romantic about it; more like emotional closeness than any sort of hormonal surge. For that matter, the person you have that sort of friendship with can have a significant other, and it really won't affect the friendship that much, because you're just really happy to see them happy. (Unless they become obsessed with their SO. That can cause problems. But then, it would cause problems for any non-romantic friendship whether or not the friend happened to be asexual.) It can happen to multiple people at the same time (the friendship I mentioned was a group of four females, actually, and the other three were straight women!), and gender doesn't matter. They don't have to be asexual; but if I were to marry someone, they would have to be asexual, and they would have to be that sort of close friend. At the moment, I don't think I'm ready to marry anyone at all, because I'm not ready to live with another person yet.
Some asexuals want romance, and have a "romantic orientation" (gay/straight/bi). I'm not one of them; like I said, if I married someone, they would be a close friend with little romance involved. Some asexuals may even fall in love with a sexual person, and have sex because that's what their partner wants. (This would be the sort of asexual who is ambivalent about sex, not actively repelled.) I'm told it's possible to maintain a relationship like that, though it takes a lot of compromise.