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. The selvage is the edge of the fabric. When you buy a piece of fabric from the roll, you get two cut edges that fray and two factory edges that look different from the rest of the fabric and don't fray. These edges are the selvage. If the grain of the fabric runs from selvage to selvage that means the natural direction of the fabric (that you should use to line up your pattern pieces) runs straight across the fabric from factory edge to factory edge.
Straight line ending in arrowheads, means "place on straight grain of fabric."
A grain of fabric refers to the direction of the threads in the fabric. Fabrics have three grains: lengthwise grain (parallel to the selvage edge), crosswise grain (perpendicular to the selvage edge), and bias grain (45-degree angle to the lengthwise or crosswise grain). Each grain behaves differently when fabric is cut or manipulated.
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.
bias
BIAS
BIAS
Cross grain binding has a little more flexibility. If your borders are cut on the straight of grain, cross grain binding is a good choice. It is probably the most common type of binding used as it is both easy to make and an economical use of fabric. Bias binding is binding that is cut at a 45 degree angle from the selvedge.
The fabric for garments is cut in a way that will create the best lay when constructed. Fabric tends to stretch along the grain, which is why abutting seams are typically both cut along the grain.
Cross grain means running across the regular grain of a piece of wood.Cross grain or crosswise grain in fabrics means that the fabric grain has more stretch than lengthwise but less than bias.
Also to add to my answer above. When fabric is cut from a bolt, the cut may not be straight. Uneven edges make it hard to check the straightness of the grain.
Bias
a line going diagonally across the grain of fabric