There are, of course, many kinds of discrimination. All of them still exist to some extent in the United States but are generally becoming less of a problem.
One example of how discrimination still exists (but at a more subtle level than it once did) comes from studies of how people react to "black" names when making hiring decisions. Various studies have been done where identical resumes are sent out, some with white-sounding names and some with black-sounding names. Typically, the resumes with white-sounding names get many more calls for interviews than those with black-sounding names. This shows that, while employers would never openly discriminate on the basis of race, they may be (at least subconsciously) discriminating.