Mermaids have been around in European stories for a really long time. Their traditional job is to lure ships toward jagged rocks by looking all beautiful and then causing the ships to wreck and sink. But over time, mermaids came to symbolize a sort of sexual virtue because men could never reach them. And that's the kind of meaning Byron is playing on when he writes: "Or say they are like virtuous mermaids, whose/ Beginnings are fair faces, ends mere fishes;—/ Not that there's not a quantity of those/ Who have a due respect for their own wishes" (12.73).
Now it's important to remember that, even while he praises the mermaids' beauty, Byron reminds us that their bottom half is just pure fish. Or in other words, our idealized versions of women might not always match the reality of what lurks under the surface—or in this case, under the water. You might think you're getting a perfect, beautiful person when you're actually getting a half-fish creature. The same goes for someone you want to date just because they're beautiful: just wait until you see the ugly fish (i.e., mean personality) that lurks beneath the surface.