Short answers, pulled from the story.
Morphology is the study of how words are formed and how they relate to one another within a language. It investigates the structure of words in terms of morphemes, the smallest units that carry meaning or grammatical function. Morphology also examines how words behave as parts of speech and how they are inflected to express categories such as number, tense, and aspect.
August Schleicher introduced the term "morphology" into linguistics in 1859. The study of word structure itself is far older: the linguist Panini formulated the rules of Sanskrit morphology in the text Ashtadhyayi, and Arabic morphological studies date back to at least 1200 CE.
Inflection produces variant forms of the same lexeme, such as generating dogs from dog. Word formation produces entirely new lexemes, such as the compound dog catcher or the derived word independent from dependent. Inflected forms are governed by syntactic agreement rules; word formation is not.
The three principal approaches are morpheme-based morphology, which uses an item-and-arrangement method; lexeme-based morphology, which uses an item-and-process method; and word-based morphology, which uses a word-and-paradigm method. Each approach suits different types of languages and handles differently the cases where a single word piece expresses multiple grammatical categories at once.
Allomorphy refers to cases in which the same grammatical distinction is expressed by alternative forms of a morpheme. English plurals illustrate this: the suffix -s is voiced in dogs, unvoiced in cats, and in dishes a vowel is inserted before the -s to avoid a sound sequence that English phonotactics do not permit.
Isolating languages such as Chinese have little to no morphology, expressing grammatical relationships through word order. Agglutinative languages such as Turkish build words from many easily separable morphemes. Fusional languages such as Latin, Greek, Pashto, and Russian bind morphemes together so that a single form can encode multiple grammatical categories simultaneously.