Short answers, pulled from the story.
A word is a basic element of language that carries meaning, can be used on its own, and is uninterruptible. Linguists have reached no consensus on a single definition, and numerous attempts to find specific criteria remain controversial. Consistent definitions exist only at separate levels of description, such as the phonological, grammatical, and orthographic levels.
A morpheme is the smallest unit of language that has a meaning, even if it cannot stand on its own, while a word is made from at least one morpheme. Roots like "rock" and "god" and affixes like "-s" and "un-" are morphemes. Words with more than one root, such as "typewriter" and "cowboys", are called compound words.
Linguists use several methods, including the potential pause test, in which a speaker repeats a sentence slowly and tends to pause at word boundaries, and the indivisibility test, in which added words tend to land at boundaries. Phonetic rules like stress placement and vowel harmony also help, as do orthographic separators such as spaces and punctuation. None of these methods is foolproof, since some languages use infixes or separable affixes.
A lexeme is an item in a speaker's internal lexicon that gathers all inflected forms of a word under one heading, so the lexeme covers both the singular teapot and the plural teapots. Dictionaries list these items as lemmas. In agglutinative languages like Turkish, it is unclear whether a lexeme should include forms such as evlerinizden, meaning 'from your houses'.
Dionysius Thrax distinguished eight categories of Ancient Greek words in the 1st century BC: noun, verb, participle, article, pronoun, preposition, adverb, and conjunction. The Latin grammarians Apollonius Dyscolus and Priscian later applied his framework to Latin, replacing the article with the interjection. In the Indian tradition, Pāṇini classified words into nominal and verbal classes based on the suffixes they take.
Leonard Bloomfield introduced the concept of "Minimal Free Forms" in 1928, defining words as the smallest meaningful units of speech that can stand by themselves. The idea links phonemes, the units of sound, to lexemes, the units of meaning. Some written words such as the and of fail this test because they make no sense on their own.
Plato analyzed words through their origins and sounds and concluded there was some connection between sound and meaning, though words change over time. John Locke wrote that words are "sensible marks of ideas" chosen by voluntary imposition rather than any natural connection. Wittgenstein moved to the view that "the meaning of a word is its use in the language."