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A half is one of two equal parts of something:

Half an orange.

The plural of half is halves, pronounced /hɑ:vz/:

She cut it into two halves.

Half in noun phrases

We can use half and half of before nouns with the definite article (the), possessives (my, your) and demonstratives (this, that):

We spent half the time talking. It was an absolute waste of time for all of us.

Why don’t you have half my chocolate?

At least half of those books can be sold.

Half of, not half, can occur with a pronoun as head of the noun phrase:

Almost half of us were not allowed to vote. It was completely unfair.

Half, not half of, is used in the pattern half a/an and is followed by a noun of measurement:

It’s at least half a kilometre to those shops.

Not: … half of a kilometre … or … half kilometre

It’ll take me half an hour so I’ll see you at the club.

Not: … half of an hour … or … half hour

Half as an adverb

We can use half as an adverb which means ‘not completely’, ‘partly’:

She’s half French and half Algerian.

The theatre wasn’t even half full.

She is not half as clever as her sister.

Half as a noun

It took them a day and a half to reach the top of the mountain.

He’s lecturing on English painting in the first half of the nineteenth century.

It looks like it’s broken but we should be able to glue the two halves together.

Half as an adjective

They’re offering the flowers for almost half price.

Children under sixteen can travel half fare on trains.

(“Half” from English Grammar Today © Cambridge University Press. Need grammar practice? Try English Grammar Today with Workbook.)
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