Handicap Placement
Here are the traditional placements for handicap stones under Japanese Rules. Other rules, such as Chinese rules, allow free handicap placement.
Andrew Grant: It is a fallacy that Japanese rules do not allow free handicap placement. In fact the official Japanese rules say nothing at all about handicap placement. It's true that the fixed pattern is universally used in Japan, but this is just a tradition, nothing more.
In all cases the weaker player plays Black, and there is no komi (or it is just 0.5, to prevent a draw). After handicap stones have been placed, it is white's turn to play.
One handicap stone
There is no specific placement for one stone, since this would be equivalent to dictating black's first move. Black plays first as in a non-handicap game, but benefits from the absence of komi.
Two handicap stones
Joshual000: I believe upper left is left open for 'polite-ness' shown by Black to White (Black's 1st move is the one held by tradition - not White's)