There are certain things such as race and gender which are protected classes. It is illegal to discriminate against someone based on these attributes in various ways including employment. [1]

Beyond this, most employment in the US is "at-will", meaning that you can be fired for any reason or no reason. [2]

Some things not explicitly listed as a protected class might still pose problems if its determined that the way in which is it enforced is discriminatory. [3]

[1] https://en.wikipedia.org/wiki/Protected_class [2] https://en.wikipedia.org/wiki/At-will_employment [3] http://navbat.com/can-you-fire-someone-for-being-too-ugly/