Wheat flours contain a protein called gluten which, in the presence of water, forms an elastic network throughout the dough. This is the stuff that gives bread doughs their rubbery consistency. The whole point of kneading bread dough, in fact, is to organize the strands of gluten running through the dough into a strong, resilient, interconnected web. It is this web of protein that will entrap the bubbles of CO2 given off by the yeast as it ferments, enabling the dough to rise. Without the gluten, the CO2 would just bubble up to the surface and be lost.
But flour vary greatly in both the quantity and quality of the gluten they contain because different strains of wheat from different regions and different growing seasons have different gluten profiles. There are times when gluten is not your friend; in a cake batter, excess gluten will create a chewy, coarse-grained cake, and in pastry doughs it will produce a tough pie crust. But for bread you want lots of strong gluten to produce a well-risen and well-shaped loaf. This is why there are special flours for special purposes: cake flour, pastry flour, bread flour, etc.
All-purpose flour is typically a blend of “hard” and “soft” wheats which will perform pretty well in most roles. It usually contains 10-12% gluten. It can be used for bread, but will tend to produce a denser, flatter loaf. Some people will add 1T extra per cup of flour when using all-purpose for bread.
Bread flours have from 12-14 percent protein. They will feel decidedly more elastic while kneading, and will give full, rounded loaves. These flours are made from hard winter wheats from northern states.
Besides the quantity, the quality of the gluten will vary. Some glutens are better at forming the elastic network than others. You can judge this for yourself by making a “gluten ball” from different flours: make a stiff dough using just water and 1/4 c of flour. Knead it until it becomes quite elastic, then continue kneading it between your fingers under a stream of water. This will wash out the starch from the flour and after a few minutes of this you will have a ball of pure gluten. By playing with this ball, stretching and folding it, you will see that some are far more resistant to tearing than others. A good bread flour will enable you to pull the gluten into a thin membrane.