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An auxiliary language (or, commonly, auxlang) is a lingua franca, that is, a language that is used as a third-party communications language when two parties cannot or will not speak each other's language. In the current world, English most often serves that function, with French, Spanish, German, Hindi/Urdu, Arabic, and Russian often serving as well, even between populations that do not normally speak these languages or any related ones.
Usually, the term applies more specifically to constructed languages, that is, languages that are specifically developed rather than evolving through use in a wider community, generally with the purpose of being politically and culturally neutral. Debates rage over the best way to accomplish this -- how simple should the grammar and phonology be, what alphabet to use, where to draw vocabulary from (and whether to use existing vocabulary or make it up) -- leading many to conclude that the entire idea of a constructed auxlang is a fool's errand.
The most famous auxlangs are Esperanto (the most popular, although highly controversial due to a notably sexist vocabulary and a pointlessly complex grammar) and Volapük (known for badly mutating words to fit a rather strange sound scheme; also known in Russia as a silly word for transliterating Cyrillic into Latin-ASCII characters, and as a Swedish word for "gibberish"). A more recent attempt at an auxlang of pure reason is known as Lojban.
An opposing faction of language constructors, who call themselves "artlangers", prefers languages like Klingon, Quenya, Na'vi, and Brithenig, where nobody has a political dog in the fight, but people just appreciate them as works of art on their own.