In linguistics, a register is a variety of a language used for a particular purpose or in a particular social setting. For example, when speaking in a formal setting contrary to an informal setting, an English speaker may be more likely to use features of prescribed grammar— like pronouncing words ending in -ing with a velar nasal instead of an alveolar nasal (e.g. "walking", not "walkin'"), choosing more formal words (e.g. father vs. dad, child vs. kid, etc.), and refraining from using contractions like ain't.
Like with other types of language variation, there tends to be a spectrum of registers...