Question: I can't remember when and where I had this discussion, but I remember being corrected when I was speaking by a stranger saying that it is never correct to say *give me half of this*; instead, the grammatically correct phrase would be *give me one half of this*. I've never been a pro at where numbers fit in with the English language, so maybe someone here could shed some light on this.

Answer: 
It is perfectly acceptable to say "give me half of that". In English, "half" in understood on its own to mean "[one of two equal parts of something](http://www.merriam-webster.com/dictionary/half "Half - Definition, Merriam-Webster Dictionary\"")".

To put it another way:

* It would make no sense to say "give me no halves of that". You would just say "give me none of that".
* It would make no sense to say "give me two halves of that". You would just say "give me all of that".

Saying "give me one half of that" is redundant. It's equivalent to saying "give me one of one of those two equal parts of that."
*Comment posted as answer - as requested*

In idiomatic usage, you would seldom say 'one half of this'. You might say 'give me one half-pound pack of sugar' but the hyphen shows that it is a different construct. You might say 'give me one third of that' (as opposed to 'two thirds of that'), but with halves, the alternatives are none and all. However, even with thirds, it would be more usual to say 'a third' than 'one third'. So, whoever 'corrected' you was actually misleading you.