Here's a Little Guide I've just wrote, hope it helps.
minimum Material Required to mate against a lone king.
B+N+K vs.K (hard)
Q+K vs. K
R+K vs. K
^ all of these endgames can be performed from almost any postion on the board (provided the peices cannot be taken and you arn't careless enough to allow stalemate)
K+P vs.K however is a lot more tricky, and not always winable.
there are 3 things which decide if this is won or drawn
1) who is to move
2) position of BOTH KINGS
3) position of the Pawn(s) *Rook pawns (a,h) especially.
Imagine this position: - WK@c2
If white is to move then White wins, the pawn will become a Queen and black can do nothing about it ...1-0.
However, If it is black to move the game continues.
1.Kd5 f6 2. Ke6 3. f7 Kxf7 1/2-1/2
with black to move, he enters the "square of the pawn", which means he can chase and eventually capture it.
In simple K+P endgames there are always "key squares" which must be controled....
if you are defending, then you must control the promotion square (*i.e being in front of it, or being close enough to capture) but also, you must prevent the enemy King from getting to key Squares.
And now imagine this position : - WK@h6
This postion is Drawn, no matter who is to move, Why? well the awnser is simple -- the Black King can control all the Key squares.
If we first Imagine White to move...
White's own King is in the way of the Pawn, which means in order to promote White King must first get out of the way
In order to black to draw, all he has to do is keep the King on the H-file, which Black can do with ease.
1. Kh7 Kf7
^ and there is no way white can get out of the way of his own pawn, because black controls all the escape squares (g6,g7,g8)
And now if we Imagine Black to move.
Now White isn't in the way of his own pawm, however, it is still drawn, Why? Because black can still control all the important squares. (h8)
2. Kg7 h6+ 3. Kh8 Kg6 4. Kg8 5. h7+ Kh8 1/2-1/2
(white next move will either lose the pawn or leave black stalemated.)