To put it simply, a lexeme is something that needs a dictionary entry. You can't figure out its meaning by breaking it into smaller pieces, so you have to look it up or figure the meaning out from context.
Let's look at a couple examples:
- Dog is a lexeme. It should be obvious you can't break it into smaller pieces and figure out its meaning that way.
- Kick the bucket is also a lexeme, even though it's made up of several words. You might try breaking it down into kick, the, and bucket, but you won't be able to figure out its meaning that way, so it needs to have its own dictionary entry.
If it helps, you can think of lexemes as entries in your own mental dictionary.
Here's how one prominent linguist defines the term lexeme:
lexeme (n.) A term used by some linguists to refer to the minimal distinctive unit in the semantic system of a language. Its original motivation was to reduce the ambiguity of the term word, which applied to orthographic/phonological, grammatical and lexical levels, and to devise a more appropriate term for use in the context of discussing a language's vocabulary. The lexeme is thus postulated as the abstract unit underlying such sets of grammatical variants as walk, walks, walking, walked, or big, bigger, biggest. Idiomatic phrases, by this definition, are also considered lexemic (e.g. kick the bucket (= 'die')). Lexemes are the units which are conventionally listed in dictionaries as separate entries.
(A Dictionary of Linguistics and Phonetics, 6th ed., Crystal 2008, p.276)