One part of something when its divided into two equal parts.
Possibly from Indo-European skel-, minus the s, with k becoming h, and with f added at the end. See What's in a word?
Images of a half
What's in a word?
By contrast to the essentially two ways of saying zero, the languages listed for zero use nine or ten different ways of saying 'half.'
The English word half may (but, may not!) be related to the Indo-European root skel-, 'to cut.' If it is, then the s disappeared into oblivion (but not in the presumed relatives, see below), the k sound became h (a common transformation, see eight), and the f (sometimes p in the relatives) appeared out of the blue. The result is a wide variety of descendents with meanings having to do with something having been cut, split, or chiseled off, including these: scalp, scale, shell, shield, skill (incisiveness!), shelf (a 'split piece of wood'), sculpture, scalpel, and cutlass.
Half is also denoted by Latin-derived semi- and Greek-derived hemi- (for other examples of Latin and Greek switching s for h see six and seven). In words derived from these roots, the meaning of the hemi- or semi- always retains the sense of 'half': hemisphere, semicolon.
Semester may feel related, but it is not. Its meaning is really 'six months': se 'six', + mes which gives us month and measure.)
French-derived demi- (as in demitasse) is also unrelated, not a variant of hemi- and semi-, but yet another idea combining dis, 'broken apart' with mid, 'being in the middle.' Descendents of demi- include medium, median, mathematical mean, middle, midpoint, intermediate, and Mediterranean.