A loom holds the warp threads. The weft threads are then woven into the warp threads in a particular pattern. These are compacted to create fabric. The area at each edge of the warp where normally the weft threads change direction is referred to as the selvedge. When viewing a piece of fabric the warp threads will run parallel to the selvedge while the weft threads will run perpendicular to the selvedge.
Once woven the fabric will normally be strongest in the warp direction. That is, because the warp threads were on the loom and under tension they normally will be the strongest threads in the fabric. As a result the cloth has a tendency to collapse, that is form folds perpendicular to that side. For this reason sleeves should be cut with the warp threads running the length of the arm. Ruffs should not be cut along the length of the fabric (along the warp). In a linen even weave fabric there should be very little stretch in the warp-wise or weft-wise directions.
When directed to lay out a pattern piece “on grain” you can use either the warp or weft grains, but warp-wise grain would normally be the best choice to minimize any unexpected stretching.
The line which is 45 degress off of the grain in either direction is the “true bias”. This is where the fabric exhibits the greatest amount of stretch when pulled. Any angle off of the grain is considered “on the bias”. Any bias pull or cut will have more stretch than fabric pulled or cut on the grain. The amount of stretch will increase as the angle approaches true bias.
Also: all bets are off when talking about knit or leather.