One aspect is that language can string together small particles to produce arbitrarily complex meanings. In contrast, most animal communication is extremely simplistic - typically one sound or gesture corresponds with one piece of meaning, and the order or number of repetitions has no additional meanings. There may be some exceptions or at least grey areas (whale song, dolphin sound+dance, bee dances, some apes and birds), but the vast majority of animals have an extremely simplistic communication scheme.
Another huge difference between animal communication and human language is that animals communicate in response to direct internal or external stimuli (hungry - > chirp, see dangerous animal -> distress call). In contrast, humans communicate about much more complex feelings, and can choose when to exteriorize their language use.
Note that humans also communicate in ways that aren't language. Our "body language", our cries of pain, baby's cries, grimaces, winking, etc are the equivalent of animal communication and are not language. Sign langauge is absolutely langue though, and you'll notice it fully ticks the two boxes I was mentioning earlier.