"Langue" can mean the human part of the body - tongue ; and it also can mean language ( in the sense of a system of comunication used by a particular country or community. I will give you few exemples in which this word is used:
Ce livre a été traduit en 20 langues : This book was translated into 20 languages
Je parle couramment 5 langues : I speak 5 languages fluently
La Langue des signes américaine est parlée par 2 millions de locuteurs : The ASL is spoken by 2 million speakers.
"Langage" may refer to the phraseology and vocabulary of a particular profession, domain or group ; the method of human comunication, the way in which a group communicates in a language, or the computing system of symbols and rules for writing programs. Here are more exemples:
Le langage corporel est important pour un maître de conférence : The body language is important to a lecturer.
Le langage juridique est très difficile à comprendre : The legal language is very difficult to understand.
Je n'aime pas le language Java : I don't like the Java language
Les jeunes utilisent un language trop familier pour parler à leurs amis : The youngsters use a very familiar language to speak to their friends.