Language is so deeply embedded in almost every aspect of the way we interact with the world that it's hard to imagine what it would be like not to have it. What if we didn't have names for things? What if we didn't have experience making statements, asking questions, or talking about things that hadn't actually happened? Would we be able to think? What would our thoughts be like?
The answer to the question of whether thought is possible without language depends on what you mean by thought. Can you experience sensations, impressions, feelings without language? Yes, and very few would argue otherwise. But there is a difference between being able to experience, say, pain or light, and possessing the concepts "pain" and "light." Most would say true thought entails having the concepts.
Many artists and scientists, in describing their own inner processes while they work, say they do not use words to solve problems, but images. The autistic author Temple Grandin, in explaining how she thinks visually rather than linguistically, says that concepts for her are collections of images. Her concept of "dog," for example, "is inextricably linked to every dog I've ever known. It's as if I have a card catalog of dogs I have seen, complete with pictures, which continually grows as I add more examples to my video library." Of course, Grandin has language, and knows how to use it, so it is hard to say how much of her thinking has been influenced by it, but it is not unimaginable—and probably likely—that there are people who lack the ability to use language and think in the way she describes.
There is also evidence that deaf people cut off from language, spoken or signed, think in sophisticated ways before they have been exposed to language. When they later learn language, they can describe the experience of having had thoughts like those of the 15 year old boy who wrote in 1836, after being educated at a school for the deaf, that he remembered thinking in his pre-language days "that perhaps the moon would strike me, and I thought that perhaps my parents were strong, and would fight the moon, and it would fail, and I mocked the moon." Also, the spontaneous sign languages developed by deaf students without language models, in places like Nicaragua, display the kind of thinking that goes far beyond mere sensory impression or practical problem solving.
However, while it appears that we can indeed think without language, it is also the case that there are certain kinds of thinking that are made possible by language. Language gives us symbols we can use to fix ideas, reflect on them and hold them up for observation. It allows for a level of abstract reasoning we wouldn't have otherwise. The philosopher Peter Carruthers has argued that there is a type of inner, explicitly linguistic thinking that allows us to bring our own thoughts into conscious awareness. We may be able to think without language, but language lets us know that we are thinking.