Defining what language is, seems, at first glance at least, pretty straight forward: Words which combined together communicate ideas to other people. Inevitably the question of wether or not animals have / or use language comes about, and when pushed the answer seems, somewhere down inside, to be no. Only people have language.
For linguists, as one can imagine, the story is much more complicated. First of all language is more than words it is also how the words are combined together, or grammar. Second animals, as it turns out, can use language, just not complicated nuanced language. For example, bees, parrots, and chimps can approximate language but not with the complexity or spontaneity that comes naturally to human beings. Also the question is still out on how language began, wether or not Neanderthals could speak in the same manner as Homo sapiens and how we developed the ability to use language in the first place.
The process began with linguists setting about trying to define the problem, first from a very Homo Sapiens-centric point of view (i.e. lets teach chimps English) and then a wider relative one. Linguists asked began to ask what language was, how it was different than non-language, what were its stages of development, its morphology and how its use shapes the way we view the world. As one can imagine the idea of language became quickly very complicated and increasingly controversial as well. Charles Hockett, for example defined language as having 13 features. http://en.wikipedia.org/wiki/Charles_F._Hockett, Only Human language has all 13. For example, #10, Displacement, or the ability to refer to things in space and time and communicate about things that are currently not present, bees and humans have but apes do not; Noam Chompsky believes that Humans have a language gene.
But dispite the studies, language still remains, for the 'lay person' something much more tangible and viseral.