Contemporary art means modern art done after Pop Art and Conceptual Art. Another term, which really means the same thing, is postmodern art. Yet another term, which goes wider than art, is Postmodernism.
This problem of labelling comes about because the word 'modern' in art was used for some art from the mid-Victorian period through the whole 20th century. It covers a wide range of styles and movements. So when art critics wanted to describe post-WWII art they had to use a different word. Thus, in art, 'contemporary' (which really means 'at the same time as') comes to mean 'modern', and modern comes to mean 'earlier'.