Many intermediate chess players know that
if it was white's turn to move he can win with
b6 and soon he will have a queen!
     However, what if black is to play?  There is
one move that will secure the draw for black
and one move that actually wins for black.

     Can you find these two moves and justify
your answer?

UPDATE: If you don't want to see the answer avoid Posts #5-8.

is that helpful? even 1..a6 is a good move.

AnandBenegal, in your variation white missed a chance to win with 3. a6.

1...Kf5 wins for black, since the b6 trick fails, i.e. 1...Kf5 2.b6 cxb6 3. a6 bxa6 4.c6 Ke6 will catch the pawn in time. 

Correct.  There is also a move for black other than Kf5 that leads to at least a draw.

1.... b6 draws:

Correct.  Also, b6 can win if white does not play correctly.