A simple definition of thick description will be an account of a culture or society which includes elaborate information on context and social background to explain behavior.
Clifford Geertz, who coined the term thick description, meant two things by it. The first practical meaning of asking an observer to write a "thick description" is to ask him not only to account for what he sees but also to try and find the context which will account for what he sees. The second theoretical meaning of Geertz's thick description is that social behavior should be understood "deeply", that it to understand that any behavior is rooted in elaborate social and cultural constraints that the researcher must bring into account if he wants to accurately portray a society.
If we take the example of the Nacirema culture, it is not enough to record women putting their heads into ovens or to describe the in-house shrines. One has to dig deeper and explain how these phenomena are related to the Nacirema's body culture (to understand the example see: Body Ritual Among the Nacirema by Horace Miner).
Greetz developed the idea of thick description in his seminal "Thick Description: Toward an Interpretive Theory of Culture". Geertz gave a brilliant examples of his theory in his famous articles " Deep Play: Notes on the Balinese Cockfight" or Person Time and Conduct in Bali . Another important theoretical article by Geertz is " From the Native's Point of View: On the Nature of Anthropological Understanding" (all these articles have summaries on this site, follow the links).
Geertz's "thick description" was and still is a highly influential line of thought in anthropology, ethnology, ethnography, sociology and all of the social sciences. Geertz also influenced to trend in anthropology that sought to understand cultures from within, and thick description is an important tool for that.