Racism is a form of discrimination, based on race. Like other forms of discrimination, the core idea is that there are generalized differences between two groups, and that those differences make one group inherently superior.
More specifically, racism is the belief that human races exist and these races possess different abilities and characteristics and that some races are superior to others, making us inherently different and unequal. Often, racists do not want to live in multi-racial societies, rather they encourage separation and segregation and want to live amongst their "own kind." Others, more in keeping with other forms of discrimination, believe that it is perfectly fine to live among members of the "other" group, as long as there is a clear understanding of who is better and who has the power. Subservience and control is usually the goal in this case, as with most gender discrimination.
Racism does not inherently include violence as part of the definition, but when one group believes they are superior to another group, then biased behaviors based on those beliefs can spawn disrespect and humiliation. In extreme cases, racist beliefs (or any discriminatory beliefs) can promote violence, defamation, murder, rape, torture, and/or genocide.