This is a very important question. Possibly THE important question for the purposes of this thread. I agree that it's time we dealt with this highly important issue.
What is a P/Cer?
This came up recently, when there was a discussion over what a fan story marked "P/C" implies, or promises. Now, I mark most of my stories with that code, but I don't know the answer, and I don't think we ever sorted it out.
At the root, I would say a P/C shipper is someone who: likes Jean-Luc Picard, likes Beverly Crusher, likes the interactions between the two of them, and wants to see more. I think it's all in how you define the "more".
Fundamentally the "more" is either romantic or sexual, but the romantic can simply involve the exploration of feelings (usually in fanfic) rather than an actual relationship. I think P/Cers range from those who wish The Couple had gotten married and had children early in the series, to those "finishippers" (term coined by the Archer/T'Pol people) who want them to get together eventually, but not necessarily now. Not to mention the fans of angst, who just want tragic unrequited love. So it's a broad church.
It's been made clear to me that people have made assumptions about my attitudes to certain things, based on the fact that I'm a P/C shipper. Actually I don't hate Nella Daren, or Vash, or Jack Crusher, or indeed think that Jean-Luc and Beverly should have started a relationship while serving on the Enterprise. At the same time, though, I don't consider that I'm lukewarm in my level of devotion to the relationship. How many posts have I made in this thread? Too many.
So there are my musings. Anyone else have any insight?