The other night I attended a gathering at our gallery and was asked the following question: "What is the difference between Contemporary Art; Modern Art and Abstract Art?" I knew that abstract art was basically art that separates itself from reality and which really took on some movement when Jackson Pollock came onto the abstract expressionism scene. Abstract art is a style of painting and not a classification of an art period since we are still creating abstract paintings today.
Modern art started around 1870 by Impressionists like Claude Monet who was recognized as one of the first modern artists. He stood out from his classical peers by experimenting with new ways of seeing, expressing new ideas and methods of painting. Technically the modern art movement ended around the 1960's and 70's when the term "postmodern" started to be used and pop art became the new thing (ie Andy Warhol).
As for contemporary art -- it's here with us right now. It's art that's being created in our lifetime or in the current moment -- no matter what the style is. Anime, digitally-altered art, graffiti, collage, encaustic, mixed-media are just a few of the art movements that surround us today and would fall under the contemporary art label.
Will we always use the word "contemporary" to describe the artwork being created in the present moment? Probably not as we'll surely find another term that will signify an end of the contemporary period and the beginning of a newer one. But for now I hope this will answer the question.