What animals can speak excluding humans?
Generally speaking, among non-human animals only birds are able to sound human words clearly. Other animals just don't have the right-shaped mouth, tongue and vocal cords.. We had a cat who very definitely said "hello" in greeting but he was unable to sound the consonants, so we would say "Hello" and he would reply "Eh-o".
However, we know that at least some parrots, cetaceans, dwarf mongooses and some monkeys have identifiable "words" with set meanings within their own vocal range. Probably many species do, but those are just the ones we've studied.
We also know that many non-human animals understand quite a lot of human speech. Parrots, crows, the brighter breeds of dog, toothed whales (including dolphins), apes an elephants are all able to learn the meanings of hundreds of human words. for some reason very little work seems to have been done on communication with elephants, even though they are highly intelligent, but parrots, crows, dogs, dolphins and apes have all shown themselves to be able to talk back using intelligible if basic language. Parrots and crows can actually produce the sounds of human speech, and the others either use giant pictorial keyboards, or hand-signing. I'm sure raccoons would be able to do so as well, if we showed them how.
Most of the conversation is at the basic level of "Give me an apple", but it can go much further than that. There is or used to be an orang utan who had seen children playing in plastic paddling pools, and she understood barter because she was paid in tokens for taking part in experiments, and when the weather was hot she saved up a heap of tokens, gave them to her handler and signed "Want to buy pool". I believe it was the same one who was able to make puns, and those puns were based on the sound of the words, not their hand-sign - so she was thinking in sounds even though she couldn't reproduce them.
The famous bonobo Kanzi not only learned signing but taught it to his family. In one occasion some bonobos started fighting, and the handler was just wondering what they were fighting about when Kanzi wandered over and told him - so we have apes not just expressing basic needs but engaging in barter, word-play and gossip.
Many species of parrot not only imitate human speech but often have a good understanding of what they're saying - the famous African Grey named Alex told his handler to "Calm down" when she was upset, and another bird reacted to a human who kept trying to get him to talk by snapping "For Christ's sake shut up!" Obviously he wouldn't know that the first part was the name of a human religious figure, but he knew the second part meant "stop making noise" and that the first part added emphasis.
I've never forgotten an article I read about 20 years ago about the ravens at the Tower of London. Some of them were able to make human words, and the journalist reported hearing a keeper hold out a treat to one of the ravens and say "This is for you" and the bird replied "That's for me" - which indicates all kinds of complex mental processes. I think it was the same bird who, when he was dying, turned to the keeper who was nursing him and said "Thank you."