The leading contenders are our closest relatives, the two species of chimpanzee and the two species of gorilla. Individuals of both types have learned to "speak" in American Sign Language and the ability to use language is generally regarded as the most powerful indicator of intelligence.
Some parrots, notably Alex the African Grey, develop rudimentary abilities to actually speak and understand spoken language, which impresses us more even if their mastery isn't on a level with the apes. (I wonder what deaf people think about that.
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Cetaceans, both the baleen whales and the "toothed whales" (dolphins and the sperm whale), clearly communicate with sound patterns that vary in complex ways we haven't learned to interpret, but they might well be sophisticated enough to qualify as language. I think you'll find that several species of primates, psittacines and cetaceans are universally regarded as the most intelligent non-humans.
The other generally accepted measure of intelligence is the use of tools. (Note that we measure other animals' intelligence by our own standards.
) Crows and a few other birds have learned to fashion sticks for digging.
We also rate animals on how well they learn the tricks that we try to teach them, especially using verbal commands, so pinnipeds (seals and sea lions), dogs, horses, pigs and a few other species get high marks. However, this biases the test in favor of animals with hierarchical social or "pack" instincts as opposed to both solitary and herd-social animals. Pack-social animals are generally the only ones with a sense of hierarchy who would, therefore, regard someone else as a leader and even bother learning to follow their commands.
You can find a lot more detail on this subject by Googling, certainly more than you'll get from the non-professional biologists and psychologists who will participate in this thread. But I doubt that you'll get any more definitive answers than this one.
After all, the way we measure intelligence quantitatively, so ranking can be established, is by administering IQ tests, and no other animal can read.