Short answers, pulled from the story.
Nikāya is a Pāli word meaning "volume" or, more broadly, "collection," "assemblage," "class," or "group." In Buddhist usage it most commonly refers to groupings of discourses in the Sutta Piṭaka of the Pāli Canon, or to monastic lineages within the Theravāda tradition.
The five nikāyas within the Sutta Piṭaka are the Dīgha Nikāya (long discourses), the Majjhima Nikāya (middle-length discourses), the Saṃyutta Nikāya (thematically linked discourses), the Aṅguttara Nikāya (discourses grouped by numerical enumerations), and the Khuddaka Nikāya (the minor collection).
Nikāya and āgama are near-synonyms: nikāya is the Pāli term used in the Pāli Canon, while āgama is the Sanskrit term used by non-Theravāda schools for their equivalent collections. The Āgamas survive mainly in Classical Tibetan and Chinese translation and correspond closely to the Pāli nikāyas.
Nikāya Buddhism is a scholarly term coined by Masatoshi Nagatomifake as a non-derogatory substitute for "Hinayana," referring to the early Buddhist schools. It covers pre-sectarian Buddhism and the early Buddhist schools, though some scholars exclude pre-sectarian Buddhism from the definition.
Theravāda monastic nikāyas can arise from royal or government patronage, from the national origin of an ordination lineage, or from disputes over caste or interpretation of the monastic code. For example, the Amarapura Nikāya of Sri Lanka emerged as a reaction to caste restrictions within the Siam Nikāya.
Myanmar uses the term gaing (or gaṇa) rather than nikāya for its monastic orders. Under a 1990 law on Sangha organizations, only nine legally recognized gaings are permitted. The largest is the Thudhamma Gaing, founded in the 1800s during the Konbaung dynasty.