Contemporary dance is such a hot term right now.  Companies have named themselves in honor of it, and many dancers aspire to become a contemporary dancer.  To me, like some of you have expressed, it is a complete fusion of dance forms because in reality what is it? In most cases it is ballet with modern undertones, a little jazzy, etc.  There are too many combinations to mention, and they have been well discussed in the comments above.  It can be danced in point shoes, ballet slippers, or barefoot.  Most companies that consider themselves contemporary ballet companies still perform in point shoes, but trade traditional ballet steps for those with a twist of more modern movements. "Pointe Magazine" recently even dedicated a whole issue to Contemporary Ballet.

In the mid 1900's George Balanchine choreographed works which came to be tagged as neoclassical ballet.  This came under some criticism for its departure from the traditional ballet shapes.  He included more angular arms, parallel feet positions, and twists of the body. Those who have seen his "black and white/leotard ballets" know what I'm talking about.  All of these shapes were derived from traditional ballet steps. Under his direction of New York City Ballet he did not give a "contemporary" class, it was a full ballet class in the Russian style.  I was told by a former dancer with the New York City Ballet that he believed with this solid foundation a dancer could move and shape their body in any way that was desired.

This makes complete sense, and the development of contemporary classes is truly new wave.  If a dancer has a firm base of training, it is the job of the choreographer to mold them.  It is my opinion, and I think the opinion of many, that they should not need to be trained and conditioned in the particular style of the steps.

However, I think new movement and choreography is always necessary to dance as it moves through the ages. I'm not sure contemporary is really a genre as much as it is just choreography.