The Latin word cunnilingus meant originally ‘one who licks a woman’s genitals’ (‘q. cunnum lingens’, L&S). The word was borrowed by English, pronounced [ˌkʌn.ɪ.ˈlɪŋ.ɡəs], in the late 17th century, also to refer to the actor, as in Latin. A couple of hundred years later, however, the word was being used in English to refer to the action, i.e. the ‘stimulation of the female genitals using the tongue or lips’ (COED). Spanish too has borrowed this word in recent times, through English, for the mentioned action, also as cunnilingus [ku.ni.ˈliŋ.ɡus], whithout a spelling modification, though cunnilinguo [ku.ni.ˈliŋ.ɡu̯o] is also attested in Spanish.
The word cunnilingus contains the root ling‑, but it is not clear if it comes from Lat. lingua ‘tongue’ or from Lat. lingĕre ‘to lick’ (principal parts: lingō, lingĕre, linxī, linctum). Most likely it was a play on both of these words, though the verb was likely what originally inspired the compound noun. (The Latin expression cunnum lingĕre ‘to lick the vulva’ is found frequently in Roman graffiti, OED.) The first part of this word comes from Lat. cŭnnus that meant ‘cunt’ (vulgar word for ‘vulva’), but also ‘woman’ and ‘female pubic hair’, from where comes the patrimonial vulgar word for the female genitalia in Spanish, coño ‘cunt’. In English, cunny [ˈkʌni], earlier spelled coney, has been a ‘coarse slang’ word for ‘the female genitals; the vulva or vagina’ since at least the late 16th century (OED), but it is not directly related to Lat. cŭnnus but rather to the Latin word cŭnīcŭlus 'rabbit', and thus to Sp. conejo 'rabbit'. As for the source of Lat. cŭnnus, there are several theories, none of which has been proven.
 The word coney, now pronounced [ˈkoʊni], attested since the 13th century, means primarily ‘rabbit’ or ‘rabbit skin’. The word comes from Old French conin, ultimately from classical Latin cŭnīcŭlus ‘rabbit’ or, derived from it, ‘a passage under ground, a hole, pit, cavity, canal, etc.’ (L&S), the source of Sp. conejo ‘rabbit’, from Gr. κόνικλος (kóniklos) or κύνικλος (kúniklos), a word that may be of Iberian origin.
There are many slang terms for cunnilingus in English, such as carpet munching, muff-diving, giving lip, lip service, and dining at the Y. Curiously, someone who performs cunnilingus has been referred to as a cunnilinguist. Other rare derived words are the adjective cunnilingual and the noun cunnilinguism the practice or habit of cunnilingus’ (WNTIU).
The analogous term for ‘stimulation of the male genitals using the tongue or lips’ is known in English as fellatio [fə.ˈleɪ̯.ʃɪ.oʊ̯]. This is a New Latin term created in 1894 by Havelock Ellis, from the stem fellat‑ of the passive participle fellatus of the Latin verb fellēre ‘to suck’ and ‘to fellate’. The Spanish equivalent is felación, an adaptation of the same New Latin word. The main slang term for ‘fellation’ in English is blowjob and in Spanish, slang terms include mamada from mamar ‘to suckle’ and chupada from chupar ‘to suck’.