As fair art thou, my bonnie lass,
So deep in luve am I:
The speaker says he is as "deep in luve" as the "bonnie lass" is fair (a word that, once upon a time, meant pretty, beautiful, or attractive).
Really, this is a fancy pants way of saying something that's not so fancy pants at all. Imagine a really hot girl or guy, and now imagine that you love that person as much as he or she is hot.
Bonnie, by the way, is a word that means beautiful or pretty (just like "fair"). It is, for the most part, a Scottish dialect word. As is lass, which just refers to a girl (although sometimes it means something like sweetheart).
This guy is one sweet talker.
And I will luve thee still, my dear,
Till a' the seas gang dry:
The speaker says he will "luve" his "bonnie lass" until all the seas dry up.
The word "a'" is a shortened form of the word "all"; this elision (the removal of letters from a word) is very common in Scots English (i.e. the form of English spoken in Scotland), but you'll see it in regular English poems, too.
"Gang" doesn't refer to a group of people; it is an old word that means "go" or "walk." Say it to yourself. Doesn't it kind of sound like "gone" or "going"?
The seas will probably never "gang dry," so the speaker seems to be saying that he will love his "lass" forever. Or at least until the apocalypse.