A reaction is said to be second order when the overall order is two. The rate of a second-order reaction may be proportional to one concentration squared,  or (more commonly) to the product of two concentrations,  As an example of the first type, the reaction  is second-order in the reactant  and zero order in the reactant CO. The observed rate is given by  and is independent of the concentration of CO.
For the rate proportional to a single concentration squared, the time dependence of the concentration is given by
The time dependence for a rate proportional to two unequal concentrations is
if the concentrations are equal, they satisfy the previous equation.
The second type includes nucleophilic addition-elimination reactions, such as the alkaline hydrolysis of ethyl acetate:
This reaction is first-order in each reactant and second-order overall: 
If the same hydrolysis reaction is catalyzed by imidazole, the rate equation becomes
The rate is first-order in one reactant (ethyl acetate), and also first-order in imidazole, which as a catalyst does not appear in the overall chemical equation.
Another well-known class of second-order reactions are the S2 (bimolecular nucleophilic substitution) reactions, such as the reaction of n-butyl bromide with sodium iodide in acetone:
This same compound can be made to undergo a bimolecular (E2) elimination reaction, another common type of second-order reaction, if the sodium iodide and acetone are replaced with sodium tert-butoxide as the salt and tert-butanol as the solvent: