This is an area currently being studied by many psychologists, as it may help them understand the basis for human emotions. One cautionary note I would offer is that such studies can assess the behavior of animals but not internal states. In other words, a scientist can observe a vole comforting its mate by licking or monkeys grooming the loser in a leadership contest but cannot actually get inside animals' heads to see how they are thinking or whether animals have "cognitive empathy," which is grounded in understanding the feelings of others.
Animals do appear to experience "emotion contagion," feeling pain in response to the pain of other animals, and will also help other animals who are injured or in distress. Rats given a choice between freeing a jailed fellow rat and a delicious treat, in a 2010 study, would help the fellow rat. Other experiments and observations have yielded evidence of many forms of behavior which people would see as grounded in empathy; however, the mental processes behind those behaviors cannot at this point be fully known.