"Reducing" in this context refers to the willful creation of a lack of oxygen in the kiln atmosphere during firing which will results in metallic oxides such as red iron rust gets "reduced" back to green, black or bluish metallic iron. Green copper oxide gets "reduced" back back to its metallic red/brown etc. Reduction firing is used to produce colored glazes at will by controlling the amount of oxygen available inside the kiln when firing and cooling down. This is accomplished either by limiting the air coming into the kiln, or by increasing the fuel.
The problem is to arrive at the right temperature and the right amount of air/smoke at the exact right moments.
Carbon-soaking is an extreme reductions process in which the atmosphere is so carbon-saturated (smoky) that the pottery, if unglazed, will turn permanently gray to black in color.
Body-reduction is a reduction early in the firing before the clay body has become too vitrified, with the result that the clay body takes a baked appearance because the excited carbon in the atmosphere reacts with the iron oxide in the clay.
Glaze reduction is done at the glaze's maturing temperature. Most of the coloring agents in glazes are a form of metal-oxide or metal-carbonate. So if a kiln is reduced when the glaze reaches a fluid state, many different colors can be achieved by the loss of a metals oxygen to the starved carbon. For example, green copper-oxide becomes red copper and iron oxide becomes various colors from bluish green to black. Thus the reduction process is a decisive factor in achieving a good copper-red glaze or a iron green "celedon" glaze. Glaze reduction should be done near the end of the firing while the glaze is still fluid until it has cooled of enough so a re-oxidation can not occur. The easiest way to achieve this would be to close the air vents and let the kiln become filled with smoke at the same time as it cools down.
See also: Oxidation firing