Bran is the hard, outer layer of whole cereal grains like oats, wheat, rice, rye, and others. It's rich in protein, iron, fiber, carbohydrates, fatty acids and other nutrients, such as B-vitamins.
Which is why, nutritionally speaking, separating the germ and bran from the grain before processing it into flour is a terrible idea. The main reason anyone ever thought to do this in the first place was the fact that the germ and the bran contain fatty acids which can go rancid. Therefore, flour made from grain that has had the bran and germ removed will theoretically have a longer shelf life.
However, the fact that plain white flour has had so many nutrients stripped out is why flour is so often "enriched" by adding vitamins back into the flour shortly before it's packaged and sold. It's unclear whether adding vitamins into the flour is as nutritionally sound as leaving them in the grains in the first place, so from a nutritional standpoint, you're better off using whole wheat flour. When it comes to keeping it fresh, heat, oxygen, and light are the enemy. To avoid rancidity, get in the habit of storing your flour tightly sealed, in a cool, dry, dark place.
Bran: It's What Makes Brown Rice Brown
Rice also has a layer of bran on the outside of each grain, which is the main difference between white rice and brown rice. Brown rice has the layer of bran intact while white rice has had it removed during the milling process.
The same nutritional issues exist with the removal of the bran from rice as in other grains. And indeed, white rice is similarly "enriched" for the same reasons.
You can purchase a bag of pure wheat bran (either at the store or online), which is one way to make a basic bran muffin (although some recipes simply call for bran cereal).