Prescriptive grammar is a set of rules created to label grammatical constructions as "correct" or "incorrect." Strictly following a language's prescriptive grammar is associated with a formal speech or writing.
The majority of a language's native speakers (even those without a formal education) have an intuitive sense of grammatical structures that sound "right." These sentences may not follow the rules of prescriptive grammar, but are widely used and understood by that language's speakers. Descriptive grammar studies these non-prescriptive constructions.
The famous quote by Winston Churchill, "This is the kind of arrant pedantry up with which I will not put," highlights how prescriptive and descriptive grammar are often at odds.
Languages vary by region and are constantly changing, with little regard to prescriptive grammar. Descriptivists believe that a language lives in the mouths of its speakers, aiming to map and study a language's evolution independent of its prescriptive decrees.
Prescriptive grammarians believe that a permissive, descriptive approach to grammar erodes a language. Prescriptive grammarians try to reign in the evolution of a language in order to standardize its use worldwide.