1) (Most common manifestation) A white person in a white society who sees non-whites as being inferior AND has the ability and/or desire to prevent non-whites from achieving success and/or quality of life within said white-controlled society. The institutions controlled by a racist society (schools, courts, jails) will necessarily be racist as well.
2) A society of the same mechanism (i.e. the group in control of the society keeps others from being successful) within another culture (i.e. India, caste system)
3) Commonly confused with prejudice, bigotry and bias, which can be exhibited by any individual in any society. Racism has a different connotation associated with the person being able to use prejudice, bigotry and bias to, for example, kill someone because they were black (and "inferior") and THEN be acquitted at a murder trial with a racist jury and a racist judge. A black person committing this crime out of prejudice, bigotry and/or bias would not be acquitted by a racist jury and a racist trial...in fact, they probably would have executed the black person and realized the cops just picked up a random black man off the street (because he was a black man), so they executed the wrong person. If it was a black-controlled society with white people systematically being executed for committing crimes they didn't commit, one could argue that that particular black society might be racist.
A racist society will most likely have a majority of non-whites employed in some sort of servitude to reinforce the position of non-whites as inferior, presently or in the past. Contrary to popular white racist belief, non-whites are not employed as maids, servants, nannies because they love serving white people. A racist society is designed so that a majority of non-whites must openly acknowledge inferiority (usually through servitude) in order to be allowed a small share of resources - typically just enough to stay alive.
A lot of people on this thread were probably called racists and decided to deflect the definition of the word back on the person who called them racist INSTEAD OF trying to understand why people are calling them racists. Please ask why and they might explain. Racism is EXPECTED...A NORM...for minorities in the US and Europe...when someone calls you a racist, they are letting you know that you are an extreme overt manifestation; perhaps discussion will result in mutual benefit...perhaps your statements were misconstrued...or perhaps discussing it will help you realize how racist you truly are...the first step is acknowledging that you have a problem.