The word "grammar" as used by linguists is not the same as it is when used by most people. Grammar, as most people view it, is actually what is known as prescriptive grammar. Prescriptive grammar is the usual stuff they teach in school having to do with what to do and not to do when speaking and writing, in other words, someone (usually the people writing the books) prescribes what people are supposed to do. Most linguists deal with descriptive grammar. Descriptive grammar deals with how language is actually used. When linguists talk about being grammatically correct, they simply mean using language in a way that is normal and understandable in the community. Therefore, many things that are often portrayed as "bad grammar," such as "ain't," are simply part of the grammar in a particular community or dialect.
Dialect is another word that has different meanings in linguistics and everyday usage. The word is commonly used to refer to non-standard varieties of a language (which always exist), and is often used in a derogatory sense. However, in linguistics, a dialect is simply any variety of a language. So, to a linguist, the Queen's English is just another dialect.
There are several branches of linguistics. Here are some of the most important.
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