I guess if you expand what the term language means you could expand it to quite a few species. Simple chemical communication between ants comes to mind, but unless you can make those same chemicals you cannot speak and you need a way to analyze the chemicals they put down to read their language. Bees perform a sort of dance to relay info about location of flowers to gather pollen. I remember an article where scientists used a fake bee puppet to communicate.
Otherwise you could read the body language and behavior of wild animals to know what they are saying at a sub basic level.
Domestic animals communication is a bit better, some people do have a knack and understanding of how dogs think. They make excellent trainers.
Corvids and a few other bird species are remarkably intelligent even showing crude tool use (look Mah no opposable thumbs). I suppose if you can simulate the bird sounds and decrypt what they mean you can listen and speak bird.
I would say, however, that animals do not have a common language. Can't speak to a raven and an ant with the same skill. The species would need to be very closely related to have a shared language, and even then they would have dialects. Look at Eastern and Western meadowlarks which sometimes interbreed to make hybrids, but have different bird songs.
Most animals would have very simple concepts. Food this way. Alert predator! Attack!.
Chimps would have more. They have been shown to be able to learn sign language and communicate with humans with a fairly broad language base. It is unclear whether these trained chimps have more of a vocabulary than wild chimps. Perhaps through their training their brains developed more for communication than wild born chimps do. ( their is some elasticity in development, not everything is genetics.)