What does grain of the fabric mean?

The way the fabric is woven gives it different properties in different directions. When you are cutting a pattern you want to get all the pieces on the same "grain", i.e. a piece that will be vertical on your body should not be cut diagonally on the peace of fabric (unless you cut all the pieces diagonally or on the bias). The grain of the fabric is the natural direction of the fabric, usually up and down along the length of the fabric.