Yeah, this is going to be a bit tricky.
- halv = the adjective half
- halva = the noun half
- hälft = also the noun half, and synonymous - but this is an archaic form, I suggest you don't use it
When you want to say "half of", you can choose between "halva" and "hälften av". For whatever reason, though hälft is archaic, hälften is not. If you feel unsure, use the former for countables in singular (e.g. "halva boken"), and the latter for countables in plural (e.g. "hälften av böckerna"), uncountables (e.g. "hälften av vattnet"), and count words (e.g. "hälften av sex", "hälften av fem böcker"). It may be perceived as slightly more correct or less awkward to a native speaker, but the truth is they are pretty much synonymous nowadays, except for when using count words.
And finally, for "[article] half", you can use either "halvan" or (again) "hälften", where the former is more common. But neither English nor Swedish uses articles in that way a lot. One example would be "one half of" (in the sense of "one of either half of the apple", not "one half of the water"), which is either "ena halvan av" or "ena hälften av"
You'll likely only ever encounter hälft in the fixed phrase äkta hälft, which means "spouse". (Here, äkta means "married" - otherwise it usually means "genuine".)
Sorry if that was confusing, it's getting a wee bit late over here.
Halva is more common than hälft (apart from hälften) in everyday language as you say, but I wouldn’t call it ’archaic’ since you still see it. Archaic means it’s no longer in use, but you find plenty of usages in newspapers for example, even though most of them are used to talk about ’half a century’ or your ’better half’ as you say.
I mean it as in en halv fisk = "one half fish". That is, it is not "one half" but rather that the fish is half. So halv is the same type of word here as if it were, for instance, the word "yellow instead": en gul fisk = "a yellow fish".
English doesn't really use "half" that way, however - typically, it would be "half a fish". So it sounds a bit awkward - but it's that way we do it in Swedish.
I am all confused here. I've read all the comments, but still unsure so please let me know if I got it right at all. It's not easy to get a hold of it through english, because it seems much simpler in english. I will remember it more easily through my native tongue if I can only translate it correctly . So basically: Det ar en ful halv fisk - that is an ugly halved fish (adjective). But: Jag koper halva/halften av den stora fisken - I buy half of the big fish. And finally Det ar en halva fisk - That is a half of the fish (noun). Help!