Half-metallic ferromagnetism was first proposed in the early 1980s in this classic theoretical paper

New Class of Materials: Half-Metallic Ferromagnets
R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J. Buschow, Phys. Rev. Lett. 50 (1983) 2024.

Half-metallicity is best explained by comparing the density of states (DOS) of a normal nonmagnetic semiconductor (or insulator), a normal metallic ferromagnet and a half-metal.

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A semiconductor has no electronic states around the Fermi level, i.e. has a true band gap. This means that only impurity states (usually due to deliberately-introduced donors or acceptors) or electrons thermally excited across the gap can contribute to electrical conduction. In contrast, metals have no band gap and lots of electrons at the Fermi level available for electrical conduction. In a ferromagnetic metal below its Curie temperature, the DOS of the valence electrons is a bit different between the two available spin states due to the magnetic coupling.

This page from Trinity College Dublin has a more technical introduction to half-metals.

What is spin polarisation?

This is typically given as the normalized difference between the spin-up and spin-down DOS at the Fermi level.

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So what's the difference between a "half-metal" and a "semi-metal"...?

The term semi-metal is normally used for materials with conductivity intermediate between a semiconductor and a metal. This means that there is no true band gap straddling the Fermi level but that the density of states is very low around the Fermi level. Often the DOS is exactly zero right at the Fermi level - for instance in undoped, pristine graphene this is the Dirac point.