The word "Nip" is mainly used in the British English, but informal, and it is meant "go quickly" or go/be somewhere for only a short time. E.g.: Shall we nip in to a café for grabbing something to eat? = Shall we go quickly to a café for eating something?

Only in BrE. This is meaningless in American English. 'doctor' not 'docter'

It means '[quickly] go to'. Synonyms in a similar register (level of language) include 'pop to'.