The Collaborative International Dictionary of English v.0.48:

Grammar \Gram"mar\, n. [OE. gramere, OF. gramaire, F. grammaire
   Prob. fr. L. gramatica Gr ?, fem. of ? skilled in grammar,
   fr. ? letter. See Gramme, Graphic, and cf. Grammatical,
   Gramarye.]
   1. The science which treats of the principles of language;
      the study of forms of speech, and their relations to one
      another; the art concerned with the right use and
      application of the rules of a language, in speaking or
      writing.
      [1913 Webster]

   Note: The whole fabric of grammar rests upon the classifying
         of words according to their function in the sentence.
         --Bain.
         [1913 Webster]

   2. The art of speaking or writing with correctness or
      according to established usage; speech considered with
      regard to the rules of a grammar.
      [1913 Webster]

            The original bad grammar and bad spelling.
                                                  --Macaulay.
      [1913 Webster]

   3. A treatise on the principles of language; a book
      containing the principles and rules for correctness in
      speaking or writing.
      [1913 Webster]

   4. treatise on the elements or principles of any science; as,
      a grammar of geography.
      [1913 Webster]

   Comparative grammar, the science which determines the
      relations of kindred languages by examining and comparing
      their grammatical forms.

   Grammar school.
      (a) A school, usually endowed, in which Latin and Greek
          grammar are taught, as also other studies preparatory
          to colleges or universities; as, the famous Rugby
          Grammar School. This use of the word is more common in
          England than in the United States.
          [1913 Webster]

                When any town shall increase to the number of a
                hundred
                families or householders, they shall set up a
                grammar school, the master thereof being able to
                instruct youth so far as they may be fitted for
                the University.                   --Mass.
                                                  Records
                                                  (1647).
      (b) In the American system of graded common schools, at
          one time the term referred to an intermediate school
          between the primary school and the high school, in
          which the principles of English grammar were taught;
          now, it is synonymous with primary school or
          elementary school, being the first school at which
          children are taught subjects required by the state
          educational laws. In different communities, the
          grammar school (primary school) may have grades 1 to
          4, 1 to 6, or 1 to 8, usually together with a
          kindergarten. Schools between the primary school and
          high school are now commonly termed middle school or
          intermediate school.
          [1913 Webster +PJC]

WordNet (r) 3.0 (2006):

elementary school
    n 1: a school for young children; usually the first 6 or 8
         grades [syn: grade school, grammar school, elementary
         school, primary school]