Grain –The lengthwise and crosswise threads—warp and weft directions—of a woven fabric. The lengthwise grain runs parallel to the selvage and the least amount of stretch, whereas the crosswise grain runs perpendicular to the selvage and has slightly more give.
This is very important to consider when constructing both articles of clothing and making a quilt. When you are using a pattern for making clothes, the pattern is marked in the direction it should be placed on the grain. If you do not lay the pattern out in the correct direction, you may wind up with a poorly fitting piece.
When you are cutting out your blocks for making a quilt top, it is very important to follow the grain. You do not want those blocks to stretch when being put together. If you do not have your fabric squared up prior to cutting, you will wind up with a bias cut. This is not ideal because as you are sewing the piece into the top it will stretch out of shape.
You also want the fabric to look uniform in your project. If you run one side of a blouse one way on the grain and the other side in the opposite way, your fabric will look slightly different in your blouse. It is always better to have more fabric so that you can lay out your pattern correctly, then to do all of the work making something and it not looking right.
Below is a great image that shows the grain of the fabric.