I'm glad to see you asking about this tarcone. I think the most important thing is this step that you're taking: trying to figure out what it is. Becoming aware of the ways in which the history, culture, political systems, etc in this country have given an advantage to white folks is an eye-opening experience.
I grew up in Seattle, a very white town that is often thought of as very liberal. It is in a lot of ways, but it (like many northern parts of the country) has had plenty of issues of racism, both overt and covert - this isn't a problem confined to the south. I had no idea until recently the redlining that was present in this city up until quite recently. If you think about how wealth can accumulate over generations in a family, property ownership is usually the biggest (by far) driver of that wealth. Not allowing, or putting up severe barriers to home ownership to minorities has had a massive financial impact on generations of minorities.
And that's just one aspect of white privilege. Here's a good primer for us white folks on trying to understand what it can be like in this country for minorities:
My White Friend Asked Me on Facebook to Explain White Privilege. I Decided to Be Honest by Lori Lakin Hutcherson — YES! Magazine
As for what to do about it? Becoming aware of your privilege is likely to make you much more sympathetic to what minorities are saying when they talk about their experiences in this country, and perhaps lead you to change your views on certain policies and how you vote. It may lead you to be more aware of your behavior and re-assess if you are viewing things through an unfair lens.
The goal isn't to tear down white folks. It's to try to build a more fair society where everyone enjoys the same level of trust, opportunity and benefit of the doubt.
(and yes, not all white folks are created equal - some experience more privilege than others; what folks are talking about with white privilege is more about the macro than the micro)