Grammar is the study of the rules governing the use of a language. That set of rules is also called the grammar of the language, and each language has its own distinct grammar. Grammar is part of the general study of language called linguistics.
Prescriptive grammar -- an attempt to tell the users of the language how to use it in order to speak correctly. This is the sense in which some people state that "I didn't do nothing" is bad grammar.
Descriptive grammar -- an attempt to describe the language as it is being used, regardless of whether it is considered correct or not. In many dialects, people say "I didn't do nothing"; a descriptive grammar of such dialects would accordingly treat that sentence as grammatical and provide rules that account for it. Likewise a descriptive grammar of formal English would provide rules accounting for "I didn't do anything."
Teaching grammar -- a combination of prescriptive and descriptive approaches with the aim of teaching a language to children and foreigners. In teaching grammars it is often necessary to simplify in order to achieve success, as neither the prescriptive nor the descriptive approaches are logical or easy to understand in all details.
Generative grammar -- A technical linguistic term. A generative grammar for a particular language specifies, for each string of words, whether or not that string constitutes a grammatical sentence in that language. It does not provide a set of rules for constructing or parsing sentences.
Descriptive grammar takes the approach that speakers of a language follow that language's grammar as a common convention of mutual intelligibility. Violation of the grammar makes one's speech difficult to understand (as in "barked dog me at time for long"). A majority of modern linguists accept that no person whose brain functions are not severely impaired speaks ungrammatically in this sense.
Grammars evolve through usage and human population separations. With the advent of written representations, formal rules about language usage tend to appear also. Formal grammars are codifications of usage that are developed by observation. As the rules become established and developed, the prescriptive concept of grammatical correctness can arise. This often creates a gulf between contemporary usage and that which is accepted as correct. Linguists normally consider that prescriptive grammars do not have any justification beyond their authors' aesthetic tastes. However, prescriptions are considered in sociolinguistics as part of the explanation for why some people say "I didn't do nothing", some say "I didn't do anything", and some say one or the other depending on social context.
The formal study of grammar is an important part of education from a young age through advanced learning, though the rules taught in schools are not a "grammar" in the sense most linguists use the term, as they are often prescriptive rather than descriptive.
Planned languages are more common in the modern day. Many have been designed to aid human communication (such as Esperanto or the intercultural, highly logic-compatible artificial language Lojban) or created as part of a work of Fiction (such as the Klingon language and Elvish language). Each of these artificial languages has its own grammar.
Programming languages used for the purpose of computer programming (such as Java) have grammars, but do not resemble human languages very much. These are called formal grammars. In particular, they conform precisely to a grammar generated by a pushdown automaton with arbitrarily complex commands. They usually lack questions, exclamations, simile, metaphor and other features of human languages.
It is a myth that analytic languages have simpler grammar than synthetic languages. That languages have different levels of grammatical complexity can be shown to be false by realizing the fact that changes to words are not the only kind of grammar. Chinese is very context dependent. In other words, context accomplishes the same role as declension and conjugation. (Chinese does have some inflections, and had more in the past.) Latin, which is synthetic, uses affixes and inflections to accomplish the same role that Chinese does with syntax. Because Latin words are quite (though not completely) self-contained, a sentence can be made from scattered elements. In short, Latin has a complex affixion and a simple syntax, while Chinese has the opposite.
