What is Oxidant, reductant, redox couple, half-equation?

An oxidant is a species able to gain electrons, a reductant a species able to give them. An oxidant Ox and the reductant Red it becomes form a redox couple Ox/Red, described by its half-equation $\alpha\,\mathrm{Ox} + n\,\mathrm{e^-} \rightleftharpoons \beta\,\mathrm{Red}$, balanced in atoms with $\ce{H2O}$ and $\ce{H+}$ when needed and in charge with the electrons.