Wikipedia has a lot of detail on
Animal communication and
Animal language. Outside of animal language, there has been a lot of research on
Animal cognition. That research has included research on perception, attention, concepts and categories, memory, spatial cognition, timing, tool and weapon use, reasoning and problem solving, cognitive bias, language, insight, numeracy, intelligence, theory of mind (
Theory of mind in animals), and consciousness (
Mirror test,
List of Animals That Have Passed the Mirror Test - Animal Cognition).
I find the mirror test most interesting. It's a test of whether one acts like one recognizes oneself in a mirror.
In our species, we typically become capable of that at around 18 to 24 months old. In Alzheimer's disease, a disease where we gradually lose mental capability, we lose that ability in the late 2nd to 3rd stage of the usual 3 stages of it.
Of our closest relatives, the great apes (chimps, gorillas, orangutans) have it, though not necessarily very consistently, but lesser apes (gibbons), Old World monkeys (macaques, baboons, colobus monkey), and New World monkeys (capuchin monkey), do not.
Elephants have it, as do the two most studied delphinids: the bottlenose dolphin and the killer whale.
Eurasian magpies (
Pica pica) have it, but other corvids tested (New Caledonian crow, jackdaw) don't.