Ain: a water source used once as a marker on the far northeastern frontier of the promised land, on the line that runs down past Riblah toward the Sea of Galilee
The name is not an invented label. It is the ordinary Hebrew word for a spring of water, which is also the word for an eye (עַיִן). Naming a place for its water was completely normal, and this same border description does it twice: two verses earlier the line passes a settlement called Hazar-enan (חֲצַר עֵינוֹן), whose name is built on the same word.
One detail makes this appearance unusual. Hebrew puts "the" in front of the word here, and names almost never take "the". So the sentence can be read two ways: the border ran east of a place called Ain, or it simply ran east of the spring, with no settlement involved at all. Readers have gone both ways, and the choice decides whether the verse names a town or points at a water source.
Where it was, nobody knows. The border is traced from Shepham down past Riblah, then east of this point, and finally down to the eastern shoulder of the Sea of Galilee, which puts it somewhere in the well-watered country north or northeast of the lake. No site has ever been matched to it, and this is the only verse in the Bible that names it.
This is a wholly different place from the small southern town that carries the same name. Nothing links the two but the spelling.