Through teaching and multiple interactions with colleagues, I've found the best answer to this question of what anthropology is involves understanding a little about how anthropology came about. Anthropology is essentially a natural science discipline. It came about at a time when many natural history type disciplines arose. As we discovered the incredible diversity of natural life in the 18th and 19th centuries, scientists began to specialize on particular taxonomic groups of related organisms. Thus we started to have mammalogists and herpetologists and so forth. After Darwin we had a fully viable mechanism for how all the diversity of life could be linked together in a single unbroken and absolutely continuous history. Anthropology was born of the realization that this same unbroken character of evolutionary history must apply to humans; that all the incredible biological and cultural diversity of our species arose from a less diverse origin, and that it came about through evolutionary processes. Since we are humans, it seemed reasonable that there should be a natural history science about ourselves. Hence anthropology.
Because anthropology was born of the mindset of naturalists pursuing science, it made sense to early anthropologists like Edward Tylor and Louis H. Morgan that anthropology would pursue both a survey of extant cultural diversity and would investigate the archeological and fossil record of human existence as part of one discipline. This is, after all, exactly how a natural historian of the time would attempt to understand the evolutionary diversification of a related group of fish or rodents. You would want to know the existing diversity, about which you can of course have much more detailed information, but then also be linking that as much as possible to the direct evidence of past evolutionary change from the fossil record. Comparison and comparative methods have always been key to the study of evolution in any set of organisms. Indeed, comparison across many geographic scales, comparison among extant species, and comparison to the fossil record were all key sources of evidence for Darwin's insights on natural selection and descent with modification of species.
Given the role of language in human social life, and it's magnificent inheritance properties, it was sensible that linguistics would be brought into anthropological science at least in part. The addition of primatology to the field was also a logical broadening of the comparative basis for understanding our species' evolution.
This is what anthropology was founded to do: to be the natural history science of humankind. Such an endeavor does not encompass the study of all of human life, and the sciences of sociology and psychology had very different origins. I will touch briefly on sociology, which is often the most difficult to disambiguate from anthropology. In contrast to anthropology, sociology was not founded with the fundamental goal of understanding how human social life diversified to what it is today from a series of past mechanistic causes (evolution). Sociology established itself as strictly the science of social causes for human social life and behavior. Thus, sociology studied a type of causation that exists at a particular emergent level, just as chemists study the causation of interactions among atoms, and community ecologists study causal interactions among species, etc. Sociology even today tends to model human social interactions as analogous to particle interactions of physics and with little interest in reducing causal sequences to psychology, biology or physics through a chain of causation; rather, the social causes themselves are of primary interest. Sociology was always a level of analysis type science, and in that sense more similar to much of modern science (E. O. Wilson has written well on the contrast of natural history science and science practiced at a single emergent level).
So, that is why anthropology made sense as a distinct discipline. Does this characterize anthropology today? Not really. There are anthropologists (like me) who still are motivated principally by this vision of anthropology as the natural history science of humanity. I think there has always been at least a small core of anthropologists with this view throughout its 100-150 year existence as a distinct discipline. However, ever since Franz Boas, many and perhaps most anthropologists have not seen anthropology in this way. It was Boas who first made popular within anthropology the idea that cultural diversity just springs spontaneously from peoples' heads, and that this construction of culture by ourselves had scarcely anything to do with our biological heritage. Once anthropologists accepted that, the linkage of fossil diggers, archeologists, and ethnographers in one discipline started to seem incoherent. This was exacerbated by the increasing popularity of nonscientific methods of cultural analysis that rejected any reconstruction of historical diversification and rejected quantitative methods. Most of Boas' highly influential students, like Margaret Meade, helped push the discipline in this direction, further splintering anthropology.
What will happen now to anthropology? I'm not sure, but I am sure that the comparative naturalist science of humanity will be conducted, whether by anthropologists, or some other group of researchers like cultural neuroscientists or psychologists. This is because there is a real academic discipline at the heart of anthropology. As we discover more and more about how genes affect our behavior, and even which genetic changes are responsible for our impressive cultural capabilities, the Boasian wall of separation between biological and cultural evolution will become more and more obviously false. So, someone will take up the effort, because there is a lot of science still to do.