(Important Note: the giver is not required to take these additional preferences about setting, era, tone, etc. into account, but it might help them come up with ideas, so include as many ideas as you like)
I love believeable characterisation, romance, stories that take the reader through a rollercoater-ride of emotions and end happily (or at the very least, hopefully), and hot, consensual sex. I'm also drawn to 'missing moments,' backstory, and the ways in which characters' pasts might shape their future. I tend to prefer 'world-compliant' fic to AUs, but would not mind at all if you chose to ignore/reverse/circumvent any of the deaths in canon, or disregard the epilogue.
In terms of pairings, I'd be very happy if both of the characters involved were from the 'beholder' list. For example, I'd love to see Slughorn/Dumbledore, or maybe Scrimgeour/Fudge or... well, there are so many intriguing possibilities! Age-gaps don't bother me, as long as there isn't extreme chan, and unlikely couples are marvellous where the writer can convince that they work within the context of the story. I'd be perfectly happy with a story involving a character who gets very little mention in the books (e.g. one of the teachers we don't really get to see), if you would like to effectively create that character yourself.
Regarding non- 'beholder' characters, Snape is my favourite, and I also have a soft-spot for Gildery Lockhart. I enjoy teacher/student or other May/December pairings as long as there is no abuse of power - or indeed if the character in the ostensibly 'weaker' position is in truth the most in control. I'm much more interested in adult characters than Harry's generation of characters (and the epilogue children don't really interest me at all). I prefer 'good' or ambiguous characters to those who are portrayed as straightforwardly evil/nasty in canon. </b>
On that note, Slughorn is one of my favourite characters; I see him as someone who is fundamentally honourable and moral, but is also flawed in a lot of different and interesting ways. Similarly, my take on Dumbledore is as a basically good person. He is far from perfect - that we know - but I tend to find evil!manipulative!nasty!Dumbledore too far in the opposite direction. Rufus also is fundamentally a 'good guy' in my books, despite his flaws.