Comparative method is about looking at an object of study in relation to another. The object of study is normally compared across space and/or time. Comparative methods can be qualitative and quantitative. Often, there is a trade-off: the more cases to compare, the less comparable variables available and vice versa.
The comparative method is often applied when looking for patterns of similarities and differences, explaining continuity and change. Often applied in comparative research is the Most Similar Systems Design (that consists in comparing very similar cases that differ in the dependent variable, on the assumption that this will make it easier to find those independent variables which explain the presence/absence of the dependent variable) or the Most Different Systems Design (comparing very different cases, all of which have the same dependent variable in common, so that any other circumstance that is present in all the cases can be regarded as the independent variable).
A challenge in comparative research is that what may seem as the same category across countries may in fact be defined very differently in these same countries.