Flour provides the structure for the product. The gluten, or protein, in flour, combines to form a web that traps air bubbles and sets. Starch in flour sets as it heats to add to and support the structure. In yeast breads, we want a lot of gluten formation, since it forms a stretchy web that traps carbon dioxide and steam during baking, to give bread its texture (also known as 'crumb'). Fats and sugars help prevent gluten formation. There is some simple sugar available in flour, which feeds the yeast. So if you have a bread recipe with no sugar source, that's okay - the yeast will have enough to 'eat' from the flour. The rising times will just be longer.
Bread flour is high protein flour, and produces bread that has a higher volume because it contains more stretchy gluten. Loaves made with bread flour rest for 10-15 minutes after rising before shaping the loaves so the gluten relaxes a bit and the dough is easier to work. All-purpose flour works just fine for most breads. Whole grain flours do not have as much gluten because there are other ingredients like the bran and germ which get between the gluten molecules. Whole grain flours are usually combined with bread or all-purpose flour to make a better crumb.
Fat coats gluten molecules so they can't combine as easily, contributing to the finished product's tenderness. Yeast breads that have a high proportion of fat to flour are much more tender, don't rise as high, and have a very tender mouth-feel. Fat also contributes flavor to the bread, and helps the bread brown while baking.
Sugar adds sweetness, as well as contributing to the product's browning. The main role for sugar in yeast breads is to provide food for the yeast. As the yeast grows and multiplies, it uses the sugar, forming byproducts of carbon dioxide and alcohol, which give bread its characteristic flavor. Sugar tenderizes bread by preventing the gluten from forming. Sugar also holds moisture in the finished product.
Eggs are a leavening agent and the yolks add fat for a tender and light texture. The yolks also act as an emulsifier for a smooth and even texture in the finished product. When lots of eggs are used, they contribute to the flavor of the finished product.
Liquid helps carry flavorings throughout the product, forms gluten bonds, and reacts with the starch in the protein for a strong but light structure. Liquids also act as steam during baking, contributing to the tenderness of the product. Yeast needs liquid in order to develop, reproduce, multiply, and form byproducts which make the bread rise.
Salt strengthens gluten, and adds flavor. Salt enhances flavors. In yeast breads, salt helps moderate the effect of the yeast so the bread doesn't rise too quickly.
Yeast is a one-celled plant, available in dried form, instant blend, and live cakes. In yeast bread, yeast multiplies and grows by using available sugars and water, giving off carbon dioxide and ethyl alcohol (fermentation). As long as air is available, the yeast multiplies.
In bread recipes where the bread rises for a second time, you are told to 'punch down' the dough. This breaks up small clusters or colonies of yeast cells so they can get in contact with more air and food, which is why the second rise is usually shorter than the first rise.
When I can find live cakes, I like to use them because I think the flavor is better. However, cake yeast spoils very quickly, so I try to use it within a day of buying it. You can freeze cake yeast. My second choice is active dry yeast, which I feel has a better flavor than instant-rise. Instant-rise yeast has been genetically modified and is packaged with its own food supply, because it rehydrates and becomes active instantly when mixed with liquid. This type of yeast is very convenient, but because the rise is so fast, not much flavor develops from the fermentation process.
Sourdough breads depend on yeast and bacteria starter (mixture of flour, yeast, liquid, bacteria) to provide the special sour flavor. The bacteria lowers the pH of the bread mixture, which adds to the flavor. Since the bread is more acidic (lower pH), this bread keeps longer than ordinary yeast bread. You can make starter in your own kitchen without adding any yeast if you do a lot of yeast bread baking, because the yeast cells are present in your kitchen. If you're new to working with yeast, however, add yeast to your starter.
And here's an interesting point: San Francisco Sourdough bread can only be made in San Fransisco! Scientists discovered that the bacteria in the bread was original to the area, and a wild yeast native to San Francisco was the only type that would grow with the special bacteria. Mixes are now made in that city and shipped to other parts of the country so you can make San Francisco sourdough in your home, but that special bacteria and yeast will not grow in your home kitchen, as they do for ordinary sourdough starters.