Grain is simply the direction the grass is laying. If all grass stood straight up only the slope would influence the break of putts on the green. However, grass always grows toward the setting sun and it will tend to lay in the direction of the primary water drainage.

On most golf courses these two influences will counterbalance each other so the grain will not influence the putts very much. But, at some courses the setting sun and the water drainage are the same direction so when the ball is rolling slowly it will try to follow the grain. This makes for some very unusual behaving putts.

If your putt is downhill and downgrain it will roll like it is on a linoleum floor. To compensate for the extra roll you can do one of two things.
1. Pretend the hole is short of where it actually is.
2. Strike the ball towards the the toe of the putter instead of on the sweet spot.

Whichever of these two options feels more comfortable use it and stick with it. Do not attempt to change your putting stroke by decelerating to lessen the speed. Deceleration causes both direction and speed control problems. No matter how short or fast the putt you must accelerate the putterhead through the ball to the target if you want to be successful. If the ball is going too far shorten the backstroke or try one of the recommendations above.

Good Luck!