The word "grammar" is used in dozens of ways. One, which linguists either don't like or get seriously upset about is the normative (prescriptive) sense of grammar which tells you that you can't end sentences with a preposition, and that kind of stuff. The scientific senses of "grammar" are about languages "as they really are", not how some academy wishes them to be.
You can find book in the library that tell you about the structure of languages like English, Norwegian, Tamil, Chinese and so on. Some of these are pedagogical grammars and try to teach you heuristics so that you can learn the grammar, which are a kind of descriptive grammar for a narrow purpose. You can also find non-teaching grammars which are reference works, that describe the structure of a language. Traditionally, these are not highly technical and bound to a particular theory of the nature of language, and they basically describe the kinds of sentences that you can make in the language.
The theory of generative grammar holds that there is a specific mental ability to use language, so in generative grammar (now we are talking about that book) a "grammar" is the mental system of rules that determine the structure of a language (as might be represented in a traditional descriptive grammar). Then of course one could write up a descriptive model of a language that is based on a theory of what that mental grammar looks like.
One other confusing point is that generative grammar say that the shape of a mental grammar is the result of learning based on exposure to a language, plus some built-in system that says what a "language rule" could be. For example, language rules don't require you to perform multiplication. This built-in bias is generally known as "universal grammar".
Any individual mental grammar has a syntax, semantics, phonology, phonetics and morphology (at least in one version of the theory). The (mental) grammars of languages are unique in each of these ways, at least when we are talking about "significant" language differences (English versus Dutch). When dealing with small differences in dialects (Seattle English versus Enumclaw English) the differences may be limited to one area (phonetics). "English grammar" would thus refer to any kind of English grammar including prescriptive grammar, and "Mental Grammar" could be "The theory of Mental Grammar" (universal grammar) or "any mental grammar, including English, Norwegian, Tamil...". Since there isn't just a single theory of generative grammar, there are actually many models of English mental grammar. What is not totally clear is whether my brother and I have identical mental grammars (I don't think we produce different languages, though we have different styles). It is possible that there is randomness in the mental object that children learn from exposure to their language.