Grammar is the study of words, how they are used in sentences, and how they change in different situations. The Ancient Greeks used to call it grammatikē tékhnē, the craft of letters. It can have any of these meanings:

1. The study of a language: how it works, and everything about it. This is background research on language.
2. The study of sentence structure. Rules and examples show how the language should be used. This is a correct usage grammar, as in a textbook or manual/guide.
3. The system which people learn as they grow up. This is the native-speaker's grammar.

When we speak, we use the native person's grammar, or as near as we can. When we write, we try to write with correct grammar. So, speaking and writing a language each have their own style.