I never paid much attention to the bias either until I sewed a dress and DID NOT cut one piece on the bias. How did I find out? Well, the right half of the skirt did not lay nicely - it bonked into my knee with every step I took.
so now I do pay attention. I cut my fabric (usually) across the grain - from selvage to selvage. That's the easy one to identify because the selvage is easy to see. The widest your piece of fabric can be is approximately 45" (or however wide the fabric is)
Some people like to cut their borders as one piece "WITH" the grain, so they cut ALONG the selvage. To do that, you need to buy enough yardage so you can cut however long you want the piece to be.
The bias is where the fabric is THE STRETCHIEST. It is helpful when binding a curve. For straight sewing, it comes into play whith setting triangles. It's very important to handle the bias edge with great care because it can be distorted very easily.
The bias is a very necessary concept to understand when you sew chenille, IT MUST be sewn on the bias or it won't fray properly.