Lengua means tongue but it also means language in some cases.
Lenguaje could be translated as language but i think its most accurate translation is speech.
Idioma is the best Spanish word for language. (not to be confused with "idiom" which is translated as "modismo")
Enseñame la lengua = Show me the tongue
Habla cuatro lenguas = He/She speaks four languages
Tienes que mejorar tu lenguaje = You have to improve your speech
Habla cuatro idiomas = He/She speaks four languages