Words are the units that speakers store in a mental lexicon, and morphology asks what they are made of: the first layer of the stack of structures that deals with meaning rather than sound alone. The basic answer is the morpheme, the smallest unit that carries meaning or function. A morph is a form whose function is not yet determined. A morpheme is the class of morphs that share a function, and an allomorph is one variant of that class, as with the several German plural endings.
Morphemes are sorted along several dimensions. Lexical morphemes carry the core meaning and belong to open classes that accept new members. Grammatical morphemes express categories such as number, case, and tense and belong to closed classes. Free morphemes can stand alone, bound morphemes need a host, and bound ones include prefixes, suffixes, circumfixes, or infixes. The theory even allows abstract entities, such as a zero morpheme, where a regular rule is easier to state if an empty marker is assumed. At the margins sit morphemes with unusually restricted distribution: unique morphemes occur bound to a single other morpheme (Schornstein), blocked morphemes or confixes combine with only a small, closed set of hosts (Schwieger-, Stief-), and compounding regularly inserts linking elements (Fugenelemente, like -s-, -er- or -n-) between two roots (Geburtsdatum, Kinderliteratur).
The central distinction is between inflection and derivation. Inflection produces the forms of one word, whereas word formation creates new lexemes through composition, derivation, conversion, clipping and blending. A compound has a head that determines its category and usually its meaning, while a Blauhelm is not a kind of helmet at all, which is why it counts as exocentric. Morphology thus shows that the word is not the bottom of the system but an intermediate level between sound and sentence. Once words can be inflected and built, the next question is how they combine into phrases and sentences, which is the domain of syntax.