Simply explained, culture industry is a term used by social thinkers Theodor Adorno and Max Horkheimer to describe how popular culture in the capitalist society functions like an industry in producing standardized products which produce standardized people.
A more advanced definition of culture industry draws on the seeming contradiction between human culture and mechanical industry. This is exactly Adorno and Horkheimer's point in "Culture Industry: Enlightenment as Mass Deception" (which is a chapter inside "Dialectic of Enlightenment"). They argue that culture industry is associated with late capitalism in which all forms of culture (from literature, through films and all the way to elevator music) become part of the capitalist system of production which also has deep cultural mechanisms and not just economical ones. According to Adorno and Horkheimer these cultural products are not only meant for profit (appealing to the lowest common denominator) but also produce consumers that are adapted to the needs of the capitalist system.
A simplified example which can help explain culture industry is TV lifestyles. Ever noticed how characters on TV shows you watch usually have great homes and nice cloths (except in the case in which the character is poor)? According to Adorno and Horkheimer this is not a coincidence since it's not only nice to watch good looking people leading a good looking life, these shows also send a consumerist message about how good lives should look, prompting people to adopt a certain version of the American Dream.
The concept of culture industry become widely held in sociology, media studies and critical theory and it remains functional till this days in describing how mass culture and big business are inherently bound together to make up a large scale system of control and exploitation. Some years after "Culture Industry: Enlightenment as Mass Deception" Adorno wrote another article titled "Culture Industry Reconsidered" (see link for a summary), elaborating on the ideas and definitions of his initial essay with Horkheimer.
See also: One Dimensional man / Herbert Marcuse