As a mod, I like making applications that really have you delve into what the character is like, their history, and what makes them tick. I like to see strengths and weaknesses, and love when they're weighted heavier in the weaknesses section. I like to sometimes include a section for quirks or secrets, and hope that they come out in game. I ask for experience/writings maples to see what the player writes like, but I most what to know who the character is and why.
As a player, what I like about others' applications are the ones that make you answer questions as if you were your character. I like writing the ones that make me know my character, but still leave him or her room for growth and change in the game.