Privilege is a concept used in anti-racist, anti-sexist, and similar anti-oppression movements.
Anti-oppressionists use "privilege" to describe a set of advantages (or lack of disadvantages) enjoyed by a majority group, who are usually unaware of the privilege they possess. It is a term of art that may not align particularly well with the general-use word "privilege" or the programming term "privilege".
A privileged person is not necessarily prejudiced (sexist, racist, etc) as an individual, but may be part of a broader pattern of *-ism even though unaware of it.
One tool for understanding privilege is the Privilege checklist.
Other tools for revealing privilege to privileged allies who still have difficulty perceiving it are stories, such as Spot The Question.
Common phrases used to help remind people of their privilege are "check your privilege" or "your privilege is showing," A more verbose alternative is "your perspective as a _______ might mean that you're unaware that other groups struggle with issues like _______."
Many people, when asked to check their privilege, respond with "So? Am I meant to feel guilty? I didn't choose to be white/male/whatever." A good article addressing this is "Check my what?" A more constructive response than guilt is to listen carefully, learn more about perspectives and experience of people who do not have privilege on that axis, and use the leverage of privilege to amplify the perspectives and experiences of people without that privilege.
Another common response is "Just because I'm white/male/whatever doesn't mean I've had it easy." Possessing "privilege" in the anti-oppression meaning is not intended to imply that life is objectively easy, just that on that particular axis of experience it is likelier to have been easier than a person similarly situated but without that particular privilege. For example, two cisgender, neurotypical women of similar levels of poverty might be treated differently due to race. The woman who is treated slightly better due to (conscious or unconscious) bias in favor of her race would then be said to possess that privilege, even though overall she is still treated poorly.
Furthermore, passing privilege -- which would apply to people who appear to be white or male (for example) but have a very difficult life -- still confers systematic advantages, regardless of that individual's actual personal history, characteristics, or power.
A person who experiences lack of privilege on more than one axis would also experience intersectionality.