[Nota bene that the Latin word root lex means “word, law, reading,” in other words, language and its uses.]
“Lexeme: 1. A word considered as a lexical unit, in abstraction from the specific forms it takes in specific constructions, e.g. the verb ‘sing’ or ‘to sing,’ in abstraction from the varying word forms sing, sings, sang, sung, singing. Compare lemma. 2. Any other unit, e.g. a morpheme, seen has having lexical rather than grammatical meaning.
Lexical: 1. Assigned to, or involving units assigned to, a lexicon. Thus a lexical entry is an entry in the lexicon; a lexical item or lexical unit may be any word, etc. which has such an entry; rules are lexically governed if they apply only to structures including certain lexical units. 2. Specifically of words etc. distinguished as having a lexical as opposed to a grammatical meaning, or to members of a lexical as opposed to a functional category.
Lexicon: An aspect of language, or part of a linguist’s account of language, that is centered on units that have individual meanings. Distinguished as such from grammar or syntax as concerned with structures in the abstract. But structures in grammar themselves reflect the properties of the lexical units that enter into them, which may be very general or very specific. Therefore the precise scope of a lexicon, as a description of the properties of or assigned to individual units, will vary from one theory of language to another. In one account, it has been a simple subcomponent of a generative grammar, in others the basis, in itself, for most if not all specific grammatical patterns; in some an unstructured list, in others an elaborate network of entries related by lexical rules, and so on.
Usually distinguished as a theoretical concept, from a dictionary, as part of a practical description: hence e.g. a posited mental lexicon, not ‘mental dictionary.’”
Excerpted from: Matthews, P.H. The Oxford Concise Dictionary of Linguistics. New York: Oxford University Press, 2014.