There’s a tremendous variety in the levels of intelligence and, alternately, cleverness, in various animals. This often translates directly to the complexity of their language. Same goes for pack behaviours: some are learned (e.g. lions hunting), some are purely instinctual (e.g. division of labour in termite nests).
And it’s important to realize that language is not limited to vocal elements. Body languages, smells/pheromones (particularly during mating seasons) and dances often contribute to communication between animals.
From available evidence it would appear that no form of animal communication comes anywhere near the complexity of human language.
Most insects don’t really communicate beyond the instinctual level. There’s no actual thought behind it, it’s just automatic. One of the more sophisticated communications is bees “dancing” the direction and distance of fields of flowers when they arrive at their hive.
Reptiles and amphibians have vocalizations that mostly announce their presence, although frogs can get into some pretty loud “hey this is my territory” duels during summer nights at the pond.
Fish are mostly instinct-type communicators. Some use visual signals and water pressure changes in schools, helping them to coordinate movement. A few vocalize. Squid and octopi can send visual signals to each other through changing the colour of their skin; some are thought to be as smart as dogs.
Birds are near the top of the list, with highly intelligent gray parrots being taught huge vocabularies of human words to the point where some can make their own sentences. As superb mimics, they have the advantage of being able to actually shape their communications to exactly match our own. Many other species have more limited vocabularies that they use among themselves, with word count ranging from a few basic calls for nighthawks to a high number of different coordinating communications for crows.
Then there’s mammals, and most use just basic words like wolf howls or happy barks to communicate. Some are even dedicated to crossing species; cats meow at humans but not so much when we’re not around. Cetaceans like dolphins and orcas and whales have very complex songs though, and a lot of fairly complex communication goes on between them. There’s a lot of non-verbal communication that happens in chimp tribes.
Wolves certainly do use communication to coordinate their actions. It’s not just vocal, but body language as well. They even appear to vote: if their leader wants to, say, go hunting, they might gather and then cast their votes by sneezing.
That’s a very sophisticated form of animal communication, but human language still knocks it clean out of the water. We can talk about abstract things, we can talk about things happening in different places or at different times, we can even talk about things that haven’t happened yet or may never happen at all. By contrast, animals appear only to be able to communicate things that are happening here and now.
Another feature of human language is that it is “open-ended” and “productive”. This means that although we only have a finite number of symbols (sounds and gestures that make up the language), we can talk about an infinite number of things, including things nobody has ever talked about before. Animals by contrast can only talk about a finite number of things: they can say, “I am hungry,” but they can’t say, “A rose by any other name would smell as sweet.”
There are claims – and they are very controversial claims – that primates can be taught to communicate with humans in ways that mimic human language. The most famous is Koko the gorilla, who has been taught American Sign Language. The researchers working with her claim that she can not only talk about things that happened in the past, she can even make up her own phrases (for example, signing “dirty bad toilet” as an insult). But some linguists are skeptical of these claims, saying that the signs Koko produces are so vague and poorly executed that you can’t tell what, if anything, they mean.