The idea of grammatical variation in linguistics is the idea that there can be differences in grammatical usage across populations of speakers of a given language. In other words, different people can use different grammatical constructions even though they mean to say the same things. This variation is often seen as a way to distinguish between people of various classes and ethnic groups.
For example, the paper in the utsa.edu link below looks at subject-verb agreement in "to be" verbs in English. It looks at whether people from different socioeconomic groups and ethnicities speak differently with regard to the verbs they use and the degree to which their verbs and subjects agree with one another.
The idea of grammatical variation is part of a broader commitment to the idea that there is no "wrong" or "right" when it comes to grammar. Many linguists argue that the idea of correct grammar is one that is based on social and cultural dynamics. One group will come to have the social power to define its way of speaking as "correct" and all others as deviant. The notion of grammatical variation goes against this way of thinking.