Contemporary prints refer to prints that have been made during our time. They are up-to-date and made during what we refer to as the modern age. Thus prints made by Van Gogh or by Goya would not be considered contemporary whereas those by Picasso and Warhol are.
Prints today come in a vast array of formats and types and are as varied as the contemporary art scene is. In some countries are quite traditional and follow the way things have been done in that particular country for centuries. For example printmakers in Japan pay homage to that countries tradition of fine wood block printing in much of the work that they do. A lot of more recent prints in the USA follow a different kind of "tradition", that of abstract expressionism.
Since the 1950's printmaking has been stimulated by the deliberate setting up of tertiary printmaking departments in an effort to stop certain techniques from dying out. Institutions such as the Tamarind Institute and Rutgers University in the USA have ensured that contemporary prints include a healthy dose of lithography. These organisations have not only benefited printmaking in the USA but have also reached out across the globe and have resulted, through the training of master printers, in the establishment of a number of contemporary print workshops such as The Artists' Press
Hole editions in the UK Contemporary prints often combine different print techniques and elements in ways that were not done in the past. Thus a print by Toon Verhoef (Netherlands) will include collage elements that have been printed on separately.
Erika Hibbert and Joachim Schonfeldt (South Africa) will include aspects of embossing in their work which adds detail and texture to their prints.
With the vast array of prints that are available one needs to do some research before investing in work. The main criteria should be whether you like the image and whether it is in fact a genuine hand printed limited edition print. Boundaries have blurred and techniques have expanded to the point where one print technique cannot be said to be more valuable than another. One sometimes comes across the statement that etchings are more valuable than any other type of print. This is based on outdated perceptions from the 1900's and is not true. If you are not sure of what you are looking at ask for a documentation sheet and ask for details of how the print was made. If you are still not sure ask a reputable art dealer for advice.