Seeing each other to being exclusively in a relationship..

How does that happen?
What does "seeing each other" actually mean? I haven't had much experience, but once, I sort of got back together with an ex (yes I know very bad move), and well he asked me out...and I took that to mean that we were officially in a relationship. He on the other hand thought we were just "seeing each other"...despite the fact that he made no effort whatsoever to actually come and see me.
So what does this "seeing each other" business entail? I've heard that you can see other people...which kind of strikes me as weird. I'd get confused and mixed feelings for all the diiferent guys...or I'd just mope about, wondering how I could get the guy to just be in a relationship with just me, which is what would probably happen. What can I say? I'm a one-man woman.

Also, I've heard that you can just sleep with them and just carry on "seeing each other". That scares me, and also strikes me as kinda weird, and makes me uncomfortable with this whole dating/seeing each other business.

I'm an old fashioned girl. I just want to meet someone I like, get to know them, and just be in a relationship, rather than just wait for however long to skirt around the issue and what not.
What's wrong with me?