A racist person is, as an earlier poster already suggested, someone who believes that skin color is a determinant of an individual's traits, aptitudes, beliefs, and entitlement to societal privileges. (This last element has really been brought home by our president, who clearly thinks that black people should feel included in this country's politics and prosperity as an act of conditional white permission--permission that can be withdrawn for behavior deemed insufficiently grateful or deferential, as evidenced by his resentment of kneeling athletes and his tweet about certain representatives of color. Indeed, the concept of "permission" has to seen as a racist one: so someone saying that "we gave rights" to black Americans in the '50s and '60s (or "we give them so much opportunity" today) is reinforcing a racist paradigm in which those who are white even have the authority to give or rescind rights. The "we" in the sentence gives away that the state is an agent of white power first, with concessions granted under duress, or at least with an IOU, to others.)
It is important to note that a person can be racist and not act on or publicly express their racist beliefs; and it is important to note that a person can really feel that they are not racist and yet say or do something racist. One poisonous quality of white Americans today is their inability to grasp this; to understand that we might have beliefs informed by racist assumptions that we have never had exposed to us, and, furthermore, that this exposure/revelation in the course of an interaction can actually be a good thing. Being told when we have said things that are racist can be empowering, in that it might actually encourage us to analyze our own beliefs and thus understand ourselves better. But white people don't like being told anything, and we certainly don't like being urged to reflect on our flaws. (This, in itself, is a form of racism, in that it assumes that the white perspective is the default, and, as such, cannot be challenged on legitimate grounds. But why should white people be the experts on what is racist? Why should someone who has never seen his own reflection be able to describe the color of his own eyes?)