I think of romance as the outward actions or expressions of certain feelings. You can have "romantic notions" about a variety of topics, and usually what that means is that you are caught up in the need for demonstration of feeling or an idealistic fantasy. Sometimes romance can be rather superfluous.
In relationships, I think romance is the demonstrative and expressive element. Romance accompanies the love, intimacy, and/or desire one experiences with another. When you romance someone, you make an effort to do or say things that let him or her know how you feel (or to get them in the sack).
A "romantic relationship," is obviously distinct from a platonic relationship or a purely sexual relationship. I think what separates it is that usually physical intimacy and love is there, and those feelings are expressed openly (if not openly to the public then to each other). That's the best way I can explain how I see romance, anyway, but it wasn't easy.