Well, There are many ways to explain what contemporary dance is. Post-modern dance could be the way you say it in the new continent.
One way to look at it is: Anti-technique, anti-method. Ballet is one way to understand the body and the space around you. Modern dance is another way to understand the the body and the space around you. Contemporary dance tries to englobe all the techniques, or rather have no technique: it's just a way to learn in which the improtant aspect is getting to know your body and your abilities, and gaining a perfect body mind connection so that you are able to do... anything.
Another way to look at it is that it is a way of dancing that is supposed to searve your own purposes. You learn "tools" that you can later use for what ever choreography you decide to do. The choreography wouldn't be any spacific style of dance, it would just be a form of expression, whether abstract or not.
Contemporary dance cannot be describes as a style, only as an attitude. I wouldn't call it post modern dance, because contemporary dance englobes post modern dance. Post modern dance wouldn't accept the use of "un-natural" movements or theatrical dance, while in comtemporary dance, it really doesn't matter. Often contemporary dancers use modern dance techniques, only to forget about them when the occasion calls for it. If you like, contemporary dance tries not to fall in a particular "box". This is why every contemporary dance teacher has his/her own personal method of teaching you.
Having said all that, there are traditions that still persist. For example: balance and unballenced movements, counter-weight dancing, falls, contact dance, etc. Improvasation plays an enormous part in many classes. Over all, I'd say that fluid and "textured" movements are what you look for. Some teachers teach ways of dancing in which you use the natural swing and flow of your body to make a dynamic dance.
So this is why you sometimes find your self doing movements you don't like, because the teacher is trying to teach you something about your body, and is trying to teach you a different attitude about dance.
I think people don't all agree about how to dfine contemporary dance. But generally all teachers in contemporary dance focus on knowing your body and how it moves, personal creativity, and finding the "mot juste", the perfect movement to fit the occasion.