Linguistics is the (scientific) study of natural human language. See a reasonably good page at Wikipedia.
Formal Linguistics
Linguists may deploy formal methods to understand grammatical strings of words, and to allow compositional theories of meaning based on the meanings of the subparts of sentences.
André Martinet, Éléments de linguistique générale, (Engl. transl.: Elements of General Linguistics)
Adrian Akmajian, Richard A. Demers, Ann K. Farmer, Robert M. Harnish, Linguistics, an introduction to language and communication
John Lyons, Semantics; Linguistics semantics
Bernard Comrie, Language universals and linguistic typology, syntax and morphology, 1981, 1989
Peter Ladefoged, Keith Johnson, A course in phonetics
Steven Pinker, The language instinct, 1994
Winfred P. Lehmann, Historical linguistics: an introduction
George Lakoff, Women, fire and dangerous things, 1987
Ronald Langacker, Foundations of cognitive grammar, vol. 1, 1987, vol. 2, 1991
Pieter A. M. Seuren, The logic of language, vol. II of Language from within; (vol. I: Language in cognition) Oxford University Press 2010
Category theory and linguistics
Joachim Lambek, Pregroups and natural language processing, The Mathematical Intelligencer 28 (2006), 41–48 pdf
Bob Coecke, Mehrnoosh Sadrzadeh, Stephen Clark, Mathematical foundations for a compositional distributional model of meaning, in the Lambek Festschrift, special issue of Linguistic Analysis, 2010 arxiv/1003.4394