I first heard this term about fifteen years ago.
My understanding is similar to that described in the Wikipedia article and a TechRepublic article: you work with your colleagues in a "friendly, collegiate way in which personal feelings are put aside". It includes things like doing peer reviews with mutual respect and a desire to learn, and not feeling like you "own" code, so if somebody has a suggestion or says there's a bug or needs to change it, you don't get defensive about it.
I've also thought it was largely about having an attitude that makes for good relations with other programmers with the goal of improving the code. So I haven't seen it as being incompatible with taking pride in the quality of your work or feeling regret if something you did caused your customer a problem.
However, an answer to a recent question makes me think some other programmers have different understandings about "egoless programming". So what is the correct definition? And what are its implications?