Yeast is a living fungus, Saccharomyces cerevisiae, and it is everywhere in the air. Yeast eats carbohydrates, farts carbon dioxide, pees alcohol, and gets hot.
When making pizza dough and bread, the CO2 is captured in pockets of the "sponge" of wet dough (shown above), a process called leavening (remember, Moses didn't have time to let the bread leaven so they ended up with matzohs, flat crackers). The small amounts of alcohol make the sponge smell like fermenting beer, but it evaporates during the baking process.
There are several different types of yeast. There's yeast for baking, for brewing, for making wine, and for nutrition. For my recipes, I standardize on instant. When you bring it home, store it in the refrigerator, and keep an eye on the expiration date. After a while, the cells die.
Active Dry Yeast (ADY). This is the most popular and it can usually be found in grocery stores. It is live yeast that has been dehydrated into tiny granules and it goes into dormancy. It often comes in convenient small packets. You rarely use a whole packet so I recommend you decant it into a small spice jar with a tight lid. No harm, no foul. It remains viable for up to two years. To use it most people "proof" or "bloom" it by dissolving the granules in warm water with a bit of sugar. When it starts to foam, usually within minutes, they mix it in with the dry ingredients. But if you are lazy like me, you can skip this step. You can just mix it in with the dry ingredients and the slightly warm water. The moisture therein is enough to wake up the cells.
Instant Yeast (a.k.a. Quick Rise Yeast or Fast Rise Yeast). Similar to ADY, but the granules are more porous and it reproduces faster. It also has an expiration date of about two years.
Fresh Yeast (a.k.a. Cake Yeast or Wet Yeast). Fresh from the factory, you'll find it in the refrigerator of your grocery, and you must use it soon. The expiration date is only about eight weeks after packaging. You can crumble it right into the dough.
Sourdough Yeast. This is a type of yeast that produces a particular flavor and an intriguing acidity. It is usually kept alive in a jar of sponge called "starter" or "mother" in the fridge, and you make more sponge by mixing the starter with flour, salt, and water.