The way we humans think and reason involves us using words to describe things. Even when thinking in our heads, we use words to think. This begs the question, how do animals who don't understand a language think?
Do you only think in words? Personally, I find I often think in images too. And sometimes my thoughts take a shortcut and skip all words and images and just take me straight to the answer? Michael Phelps is famous for thinking in terms of "the perfect race," a thought stream without words describing exactly how he would race from start to finish.
There are many features of words that are useful to us but not required for thought. For example, reproducability. I can think of "sad," pause for a moment, and then think of "sad" again and claim I am thinking of the same word. This is useful for discourse (its hard to develop a "language" otherwise), but for thought purposes, all that matters is that the "thought" works well enough right now. It doesn't matter if the thought has no meaning 15 seconds from now, the present is all that is needed for thought.
One may choose to take the issue down to the neural activity level, in which case there is clearly communication going on. However, that communication may not meet our [human] definition of language.
Does that mean animals don't "think?" Well that depends on your definition of thinking. I think you could answer any number of ways, depending on your preferred definition of the word. (Ironic that the thing that gets in the way of giving you a clear answer is that we have to use words to describe it, isn't it?)
One approach might be to attack the problem from a different perspective. Instead of presuming "language" must mean exactly what you believe it means right now, start from behaviors of the animal. Identify sets of behaviors which are connected in ways that are clear enough to warrant grouping them together. Some of those behavior groups may start to pick up characteristics of what you thought "language" meant. If so, you may elect to call them "behaviors caused by language."
A parting thought: body language. Virtually every animal has it. In fact, you could even make an argument that bacteria and fungii have it. It has no words.
I know someone who had a car accident and lost the ability to speak due to a head injury. She didn't lose the physical ability, but the part of her brain responsible for language was damaged. She had to learn to speak again just like a baby does, completely from the beginning. She is doing well now, but there are still plenty of words she doesn't know, but only the words, not the concepts (she is intelligent and can fluently insert a description for anything she doesn't remember the word for). What is very interesting is that her cognitive ability and intelligence didn't suffer in any observable way. A few months ago she couldn't even use conjugation or any pronouns, she used a grammar even worse than the Hulk from the comic books. However, no matter how hard it was for her to express her thoughts, it was evident that her cognitive ability was normal for her age of approximately 30 years.
This shows that language is not necessarily the same as thinking ability.
One approach is to extend the computational theory of mind, as Cort Ammon points out, and consider any form of neural activity as a from of computation. The answer becomes trivial: Animals think because thy process information using neurons. But we can do better than that.
We could reverse engineer your problem: How did language evolve? Before there was writing, language was a series of sounds. High order animals that clearly display some level of thinking also tend to communicate with sounds (Dogs, apes, dolphins and whales, etc...). Chances are their thought processes involves a bunch of sounds in their head as well. Not as precise as humans, but still precise enough that scientists now consider Orcas to have distinct cultures.
Douglas Hofstadter, in his book "I am a strange loop", has a section (or chapter) I forget, called "On souls and their sizes". In it he posits that, there isn't a specific break between animals and humans, but that instead there is a gradual increase in complexity of their abilities of self perception. The complexity of thoughts possible also followed a similar gradient. Presumably the animals would perceive themselves using the same types of symbols they perceive the world with, so that is how their thoughts are structured. Animals with an advanced sense of smell might associate smells with their mental images instead of sounds for example.
On the opposite side of the spectrum, you have Thomas Nagel, who wrote an article called "What's it like to be a bat?". He starts from the fact that we have no way of figuring out what the thoughts and emotions of bats are, because they are just too darn different from us. Based on this, he concludes that consciousness is irreducibly subjective and cannot be brought to a materialist/physicalist explanation. Ultimately, he arrives at a dualist position, meaning that for him, thoughts and consciousness require some dualistic mental substance to exist, even in animals.
It depends on what you do consider "thinking" - just now I am thinking about your question and I formulate those worlds directly in English in my head, even if English is not my born or more used language. But i think about answering the question in words, while my hands operates the keyboard independently.
I send command like "copy my thought in words on keyboard", "correct this typo" not "type letter A then type letter B then C", not in terms "move index finger little left, then push it down", not in forms "flex this muscle and release that one".
Does that mean, that writing on internet forum (with corrections) using standard keyboard and widespread text editing features of major browser does not need to "think"?
When I drive car, my body does a lot of work, like steering the wheel, shifting, braking, I look at other cars and avoid crashing to them, I look at traffic lights and signs and change my plans accordingly WITHOUT formulating it in words in my head - I can talk with friends on the way and use my language skills for totally different target. I could "remember back" what I did and formulate it in words (changed shift from 5 to 4, while braking and steering little left ...),but usually I do not do it. Does that mean, that driving from center of one city to the center of another, finding parking place there and park car does not involve "thinking"? Many really intelligent beginners would have problems even with the parking alone and would use a lot of words verbally (or just in head), many of which would be not publishable.
If you play with ball, you just "catch it", not solve ballistic equations in symbols and floating point arithmetic. You can describe it in such means later and get to better understanding what happened, or you can practice that and get better at that field. Your dog can "catch it" without special mathematics ability or "words" :)
So if you define "thinking" as "manipulating words", then animals probably cannot "think", but also driving in heavy traffic or using computers does not need "thinking"
If you change the definition to "solve problems in changing environment and with respect to previously learned things and with anticipating future development of situation" then you need not necessary "words" for such activity.
Cort Ammon has given a very good answer...and even better "fair warning."
You question is perfectly okay, but so exceedingly broad that is can reach into nearly all corners of philosophy, from definitions of consciousness and "life" to the role of language and "representation." To answer it one would have to usefully limit the definition of "word" and of "thought," which entails a huge chunk of modern philosophy.
Philosophers throughout the ages have attempted to define what distinguishes "animals" from "humans," usually settling on something called mind or soul or reason or language or symbolic systems, etc. For Descartes and his followers the line is quite distinct, and animals are far more like machines. Today, when we see machines simulating human "thinking" such boundaries blur. Moreover, research does show certain "language" skills in animals, so boundaries blur on the organic side as well.
Your question might be framed in many ways, but I think of two that could avoid an endless quest into the nature of human consciousness. First, what are the limits of nonhuman organic reactivity and generalization or "animal consciousness"? Second, what are "words"? What is the difference between a human saying "there is an eagle" and a monkey warning cry indicating "eagle!" Do both involve representation? Is it a matter of degree?
My own preferred framing is to look at all of these terms quite abstractly, as "communication." We can almost use this as a synonym for "society" or even "life." All lifeforms, even plants, must in some sense "communicate" between physical instantiations to continue and "re-present" themselves. This ghostly power of "communication" can be stored and transferred in various ways. All entail cells and membranes that must exclude some "communications" while remaining permeable to others. Human utterances appear to divide and redistribute experience in peculiar ways that slow and modify the physical continuum of "action" and "reaction" or "cause" and "effect." Animals do not appear to be so "loosely coupled."
But at this point in my late-night answer I am reminded that words induct us into an infinite regress from which healthy animals appear to be exempt.
Warning - this is not my area of professional expertise, but a lay observation.
When very young children learn a language for the first time, a concept ("book") and its corresponding sound are taught at the same time and are intricately linked, so they may not even recognize them as distinct. If they are exposed to a bi-lingual environment from the earliest age, they are forced to distinguish between the concept, and the representation. There may bs a single visual representation, but there are now multiple linguistic representations.
It is my belief that this leads to a more complex neuronal processing of language, and that this is fundamental in the ability to pick up additional languages (since it is only when you learn your second language that you have to start making this distinction). So a child that is bilingual from birth develops different wiring.
Animals that don't have an explicit language as we understand it can still have the conceptual processing layer, where concepts are associated with sounds, smells, tastes, emotions... Language is really just a cherry on top.
I wanted to provide some further reading; this thread was addressed rather explicitly in the 1970s with Jerry Fodor's Language of Thought Hypothesis...
which fed into a discussion about the "systematicity" of language and thought (loosely, the ability for thoughts or propositions to be combined according to systematic rules), and the implications of this for philosophy of mind.
In particular, this sparked a debate between those who felt that we necessarily think in a systematic language (on Fodor and Pylyshyn's side) and "connectionists" who invoke ideas such as "sub-symbolic mental representations" to avoid such a necessity.