Primates, birds, cetaceans, dogs and other species have proven able, through extensive training, to understand human words and simple sentences. And as Ed Yong explained, in some exceptional cases, such as Kanzi and Alex, they've even been able to engage in two-way communication with humans.
However, language is more than a process through which meaning is attached to words or short sentences. Language might be described as the ability to take a finite set of elements (such as words), and using a set of rules (grammar and syntax) to create infinite combinations, each of which is comprehensible. In addition, as far as we can tell, humans are the only species who can create and comprehend understandable silliness, like "Colourless green ideas sleep furiously".
This goes into Chinese "chengyu" or four-character idioms used heavily in mandarin, much like our "two birds, one stone." Like English idioms, origins vary between chengyu, but often they're old stories and have been used for ages in colloquial language.