Basically, a player has to be prepared to play endings in almost every opening and must have some idea about underlying endings in his repertoire. There are some differences between ending with one or two pieces and the ones mentioned above. First of all, there are some elements of a middle game, so necessary measures must be taken to secure the King. At the same point, King is becomes a strong piece in the ending, but it usually happens as more pieces are exchanged. King's power usually equals to that of a Knight or a Bishop.
Pawn structure is a second important point in such endings.
One has to be very careful making pawn moves, and must keep his pawn structure from been damaged. Otherwise pawns will be eventually lost. Perhaps a crucial part of successful endgame play is to make a plan and follow it. Simply put, make some goals you would like to achieve. Even more importantly, be consistent and follow those goals.