Here [X] denotes, for a reaction between liquids, gases, or solutes, the concentration of X; for a reaction taking place at a boundary it would denote something like moles per area of X. K(T) is a rate constant that depends on temperature. The exponents k' and m' depend on the [reaction mechanism]?, ie the sequence of steps (collisions) that the reaction takes place by, and in fact kinetics is one of the main ways of studying these. For a single-step reaction we would have
Here Ea is the activation energy, the energy per mole it is necessary for the molecules to have to react. Since at temperature T the molecules have energies according to a Boltzmann distribution, one can expect the proportion of collisions with energy greater than Ea to vary with e-Ea/RT. K has to do with stuff like the probability that molecules are in the right orientation, and of course dimensions.
For most multi-step reactions the rate is determined primarily by a single slow step, with preceding steps proceedingly quickly to a state of rough equilibrium. In general, concentrations of species are never determined solely by a single process, and as reactions occur the products begin to undergo the reverse reaction to some extent. Thus reactions never proceed to 100% completion but rather to a state of chemical equilibrium where every step occurs at the same rate as its reverse.