I used to think that flour was the most important ingredient in bread. I've since changed my mind. While flour (wheat or otherwise) provides the bulk, without yeast there would be no lift. Okay, you can make quick breads with baking powder, but when people think of bread, they're usually thinking of yeast bread.
"I think about what foodstuffs I'd bring along to an alien planet"
When I'm bored, sometimes I think about what foodstuffs I'd bring along to an alien planet. Of course I'd want to bring along plenty of flour to make cakes, cookies, muffins, and breads. I'd have my chemical leavenings in my suitcase, too, but what about yeast? It's a living thing. Would they let me bring that to my new homeworld, or would the alien version of customs police stop me at the border? And if I couldn't bring my own, would there be an alien equivalent that would allow me to culture a sourdough starter? Because without the help of yeast, my breads would be pretty dense.
Yeast is such a common thing here on Planet Earth that we don't give much thought to how amazing it is, and what a boon it is to bakers, brewers, and winemakers. And yeast is such a fun guy. Or, more accurately, a fungi. It converts the fermentable sugars in the dough into carbon dioxide and ethanol, and those bubbles, trapped in the matrix of gluten, are what causes bread to rise. When the dough is baked, the yeast dies but the pockets of air remain, giving the bread its unique texture.
The first yeasted breads were no doubt accidental, but records indicate that yeasted breads existed in ancient Egypt. Whether they were sourdoughs or a byproduct of beer-making isn't entirely clear. What is clear is that there were yeast-leavened breads long before anyone understood how it worked.
In the late 1700's, the Dutch started commercial sales of brewing yeast to be used in breadmaking, and by 1825, someone figured out how to remove most of the liquid to make solid blocks of yeast. By 1872, granulated yeast became available.
In the US, wild yeasts were commonly used until a commercial yeast was marketed by Charles Fleischmann in 1876. The Fleischmann company developed a granulated active dry yeast for the US military during WWII, and the Lesaffre company created instant yeast in the 1970s.
Today, home bakers have quite a number of options.
A live culture of yeast living inside a flour and water starter, it's the oldest form of passing on usable yeast from one batch of dough to the next. You can grow your own culture or buy a dried or fresh sourdough culture from a number of sources. It's a complex subject, but that discussion is for another day.
Fresh Compressed Yeast
Fresh yeast is live, moist yeast compressed into bricks. Some grocery stores sell individual cubes, or you might find larger blocks at specialty markets or buy some from a friendly bakery. One problem with fresh yeast is that it has an extremely short shelf-life, so there's no guarantee that the yeast you buy at a grocery store will be alive enough to use. When I buy larger quantities, I freeze the excess, with mixed results. I usually use about twice as much as I would have used if it would have been fresh, and I always have dry yeast on standby, just in case.
Active Dry Yeast
Active dry is readily available in most grocery stores. The yeast pellets are made up of live yeast cells surrounded by dehydrated cells and a growth medium. Most recipes require that active dry yeast be proofed first by allowing the dry cells and growth medium to rehydrate and dissolve in warm water. It has a longer shelf life than most other types of yeast, but some sources say it's more temperature-sensitive when in use.
This has smaller pellets than active dry yeast, with more live cells in comparison. Since it dissolves faster, it can be added directly to dough, but I often proof it anyway. If you bake a lot of sweet doughs, some manufacturers have a slightly different product that's designed for dough with a high sugar content.
This variation of instant yeast and it also has small pellets. It dissolves quickly and produces carbon dioxide faster, so the bread rises faster. Whether that's useful depends on your need for fast bread. Some recipes for rapid-rise yeast suggest a first rise of only 10 minutes before shaping the dough for its final rise. If you're in a hurry, similar results can be achieved by letting the bread rise in a warmer location or by adding more yeast.
Bread machine yeast
Another type of rapid-rise yeast. When making bread with a bread machine, the yeast is mixed directly with dry ingredients, and depending on how adjustable the timing is on the machine, it might also need to rise very quickly. So, for bread machine users, this variation has a legitimate purpose. I'm not entirely sure if bread machine yeast is a different type of rapid rise, or if it's just a different label. I found Fleischmann's bread machine yeast in a jar and rapid rise in packages, the ingredient list was the same, but there was no indication if the proportions were the same. The pellets seemed a slightly smaller than the instant yeast I had on hand, but that could vary by brand.
Also in the rapid category is Pizza Crust Yeast, a new offering from Fleischmann's. This yeast is different. Or more accurately, I'm not sure if the yeast itself is different, but it includes dough conditioners that make the dough more extensible and easier to roll. According to the package, there's no need to let the dough rise at all, it's just a matter of make and bake. The package also very clearly states that it's not intended for bread. I haven't verified how well it works.
In theory, active dry, instant, and rapid-rise yeast each have their own specific uses and should be used only in recipes that they are called for. In practice, however, nobody keeps every variety on hand; you can usually adapt your recipes to fit the type of yeast you have. Just remember that active dry yeast takes longer to rise than rapid-rise, and adjust your recipes accordingly.
"the thing to remember is that yeast is a living thing"
Some substittution charts recommend using a different starting amount of yeast, but the thing to remember is that yeast is a living thing, and while it's active, it's reproducing. No matter how much yeast you start with, it'll eventually reproduce to whatever level you'd like it to be. You can start with less yeast, give it food and optimal temperatures, and you'll end up with more happy gassy yeast than if you started with more yeast and treated it badly. So while it's great that there are options for people who need them, if you've got the right technique, you don't absolutely have to stock multiple types of yeast.