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." These phrases are mostly useful when the entire audience has a robust understanding of the concept and merely needs a reminder. A more verbose alternative that's more accessible to a newcomer 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 or defensiveness 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 (in fact it can be very difficult for any number of reasons), 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 might not actually be -- still confers some systematic advantages, regardless of that individual's actual personal history, characteristics, or power. "Passing" can be a complicated situation where a person is forced to choose between openness and safety. For example, a Jewish office worker listening to a colleague use a slur about Jewish people to express his disappointment with the lunch he bought might have to evaluate whether it is worth asking the colleague to stop. A closeted gay journalist from the United States working in a country where being homosexual is illegal might risk imprisonment or death if someone were to carelessly or maliciously "out" him to the local authorities.
A person who experiences lack of privilege on more than one axis would also experience intersectionality.