Race and racism are, I think, two massively complicated concepts that have been hugely simplified by the media, and people in this thread are using them in different ways.
I think maybe a better thread title would be "Everyone does racist things" or even "We exist inside a racist society, and as a result many of our actions, whether intentional or not, are racist".
The intentionality of racism is a very important point. Very few people
mean to do racist things, but the truth is, people in a privileged position within society (for the purposes of this discussion, white people) benefit from the racist structure of society every day. Every time you get on a train and don't have to worry that people think you're a terrorist. Every time you walk down a street and don't have people crossing the road to avoid you because they think you'll mug them. Every time you go to the shops and don't have people asking you where things are because they assume you work there.
When you're a white person, people see and treat you as a person. When you're a black person, people see and treat you as a black person. This is especially obvious if you look at the labelling of people in the US - terms like "African-American", "Asian-American" and even "Native American" are commonplace, but "European-American" is almost never used because, if you're white, you're seen as "American-American".
Being white gives you the freedom to be whoever or whatever you want (obviously within the confines of other societal classifications like gender, class and so on), but if you're not white, you're constantly forced to be a representative of your entire race, whether you like it or not.
This is a very simplified, exaggerated version of the forces at work behind white privelige, but the point is how people don't realize they're benefiting from - and thusly reinforcing - a massively racist system: