In "Chaser" we learned about a border collie who has mastered the names of over 1000 toys, and who also shows an understanding of syntax and semantics, as well as definitions. Chaser is able, in a sense, to communicate with her human. But it is a one-way communication, because of course, Chaser cannot speak. That is, she cannot speak our language.
Alex, an African Grey parrot, on the other hand, could participate in dialogue with his humans. He could also add and understand concepts of equivalency, and even came to understand zero up to a point. In many cases, he taught himself words and ideas; for example, calling an apple a "banerry," possibly combining banana and cherry.
In Irene Pepperberg's fascinating memoir about her work and life with Alex ("Alex & Me"), she is determined to discover if a parrot can do more than just parrot. Is there more than mimicry involved when the bird speaks? The answer is a resounding yes, of course, as Alex demonstrates repeatedly. He not only uses language for his demands ("Want nut!") but to emote ("I'm sorry!" "Calm down!" and "I love you"). And as he demonstrates, a creature with the brain the size of a shelled walnut can develop and learn at about a five-year-old human's level.