Improving the endgame: Principles I find I am weakest in the endgame, which is arguably the most important part of chess.  Usually when I reach the endgame I have an advantage, often a pawn or a "winning" pawn formation, but I am not sure how to capitalize and convert this into the win.

I was wondering, what are the key ideas to keep in mind when playing the endgame, and what can I do to improve my endgame play? 

I hope this isn't too general, but I believe it is an extremely common problem.
Here are a very few principles which are key:


Centralize/activate/use your king. In the endgame he is one of your strongest pieces, and needs to be utilized.
When you have the advantage, exchange pieces, and don't exchange pawns. Every exchange of pawns means one less potential promotion for the side with the advantage.
Passed pawns must be pushed, and outside passers are the best. An outside passed pawn can be the most distracting to an opponent's king or other pieces, keeping them out of other action.
Principle of 2 Weaknesses. One of the best way to convert an advantage into a win is to use it to induce another problem in the opponent's position, when he or she will then be forced to fight a war on two fronts. (For instance, this can be why an outside passer is so strong: while the opponent deals with it, a second weakness can be created and exploited on the other side of the board.)
Do not hurry! Here are some words from Shereshevsky, coming from his chapter with this title in his classic Endgame Strategy:


  How many endings have not been won, merely because the stronger side tried to win as quickly as possible, and neglected to make simple strengthening moves before embarking on positive action. Following the principle of "do not hurry", it is possible to battle for a win in positions with a slight but persistent advantage. ... The logic behind the "do not hurry" principle is mainly psychological. It can be especially recommended to act according to this principle when the opponent is deprived of active counterplay.
  
  Plenty more such principles can be formulated of course. These are just a few very general ones that can apply in most any endgame.




One book along the lines of your question that I cannot recommend highly enough is Rate Your Endgame by Edmar Mednis and Colin Crouch (based on Mednis' earlier Practical Endgame Lessons). It's now out of print, but if you can find a used copy, grab it. It offers its own explanation of endgame principles, and illustrates them with many examples, which are presented in a "test your play" format, where you are prompted to play one side of an ending from actual play. I got this book early on when I started "really" playing chess, and remember loving it, and learning a good deal. It is very much a general manual on how to approach the endgame, rather than being a discussion of various theoretical positions/endings. I've never read a poor opinion of this book, and when someone like Mark Dvoretsky gives something praise, it's not to be ignored.