River, romantic, aromantic, heteroromantic, homoromantic, biromantic, panromantic, etc are the emotional equivalent of the "-sexual" words.
So instead of whether the person is sexually attracted to people, it's about whether they're emotionally attracted to people. In other words, who they fall in love with. Aromantic means no falling in love. Doesn't mean other strong bonds don't exist and some do pair up or marry for other reasons (the advantage of raising a kid together for instance, or having each other's financial support, or many other options), but it's not what we traditionally think of as a relationship (of course on this forum, not much is traditional :P)
It has little to do with whether you give someone chocolates or flowers.
Grey-a people are people who are not quite asexual but not as sexual either, in that they only develop a sexual attraction to people they are in love with. As a result I imagine they can't be aromantic.