Short answers, pulled from the story.
A morpheme is the smallest meaningful constituent of a linguistic expression, especially within a word. For example, the word cats contains two morphemes, cat and the plural marker -s. The field of study devoted to morphemes is called morphology.
Free morphemes can function independently as words, such as town and dog, and can also appear within larger lexemes like town hall and doghouse. Bound morphemes appear only as parts of words, always joined with a root, as with the prefix un-. Most bound morphemes in English are affixes, meaning prefixes and suffixes.
Derivational morphemes change the semantic meaning or the part of speech of a word, as -ness turns the adjective happy into the noun happiness. Inflectional morphemes modify features like tense or number without changing the word's meaning or class, as -ed turns wait into waited. English has eight inflections.
An allomorph is a variant of a morpheme that differs in form but stays semantically similar. The English plural marker has three allomorphs, pronounced /s/ in cats, /ɪz/ or /əz/ in dishes, and /z/ in dogs, depending on the root. An allomorph relates to a morpheme as an allophone relates to a phoneme.
A zero-morpheme is a morpheme that carries semantic meaning but is not realized in speech, often written as / the empty set /. The word sheep serves as both singular and plural, with its plural analyzed as sheep plus a null plural suffix rather than a hypothetical sheeps. The intended meaning is derived from a co-occurring determiner such as some or a.
Content morphemes express concrete meaning and include nouns, adverbs, adjectives, main verbs, bound roots, and derivational affixes, such as fast and sad. Function morphemes serve grammatical roles and include prepositions, pronouns, determiners, auxiliary verbs, conjunctions, and inflectional affixes, such as the past-tense -ed. The two categories can overlap, as with over and your.