Yeast

In my opinion, how you treat your yeast before and during fermentation has the greatest effect on the quality of your beer. This is one of the big reasons starting with extract brewing is so useful. By focusing on the fermentation, "pitching" proper amounts of yeast, controlling temp, and you've figured out about 90% of how to make a great beer.

Yeast, specifically brewer's yeast or Saccharomyces cerevisiae, multiply by budding off new yeast through asexual reproduction. They are essentially copies of themselves. So much around using yeast for brewing has to do with pitching rates, and controlling the growth process. The idea is to pitch a certain number of yeast so they go through a very particular growth cycle:

During lag phase, the pitched yeast cells are equilibrating to the new environment and preparing for division, at the same time the new brewer (and experienced, for that metter) is anxiously awaiting fermentation to start. In brewing, this can be anywhere from 5 hours or less to 72 hours, depending on the culture, health and nutrients available. The most common question you hear from someone just getting into brewing is "are my yeast dead?" Yeast can get stressed from any number of things, from temperature swings, osmotic pressure changes, pitching at too low a density, etc. If you treat the yeast well and follow a few simple rules, the lag time should be very short, less than 12 hours.

Log phase is where the greatest amount of reproduction is happening. This requires 2 main things: nutrients and oxygen. During this time a population can double every 2-3 hours, so you can reach full density very quickly. During lag phase and into stationary phase, the temperature may swing from 5-10F higher than the ambient temp in the room, so it can get out of control very quickly. Though it makes sense that you pitch enough yeast to start fermentation directly, log stage is very important in building cell strength, health, as well as contributing many of the flavors to a beer so specific to that yeast type.

Stationary phase is anaerobic (no oxygen) and where most of the fermentation happens. When your airlock starts to bubble, you're into fermentation and the sugars are being converted into ethanol and CO2. While it's not recommended, you can easily track the progress of fermentation by watching the bubbling rate of the airlock. I'd generally say if the airlock is bubbling, that's good news, but if it's not, you'll have to check the gravity to see if the CO2 is getting out via some other path or the fermentation is truly stalled. More on that later.

Dry or Liquid?
Brewer's yeast comes in two forms, dry and liquid. There are specific benefits of either form, and of course disadvantages. Dry yeast is often cheaper than liquid, stays quite viable in storage and has more cells per package (good enough for a 5 gal batch), but is only available in a limited number of strains (ie, less control on flavor profile from the yeast). Beyond that, preference for one or the other are often personal preference. I prefer to use liquid yeast due to my experience with certain strains, but I started with dry yeast due to ease of use. Good beers with dry, good beers with liquid. The problem with liquid is that you are never provided an ideal amount of yeast to pitch for a batch in a package. Using a yeast starter, however, that problem is solved.

Outside of flavoring and fermentation specs, yeast are typically characterized by 4 main things:

Attenuation

Normally from around 65-80%, "apparent" attenuation states the average percentage of the malt sugar that can be converted into ethanol, depending on the fermentability of the wort. It's not a huge range, but small differences in residual sugar remaining in the brew after the yeast have finished will have an effect on the taste. For example, a yeast with low attenuation, in the 65-70% range, might ferment a 1.050 beer down to 1.018. A highly attenuative strain, in the 75-80% range might get you to 1.010 on average. Tweak the fermentability by adding more or less simple sugar (we'll cover this in mashing), and you can really dry out a beer or keep it very sweet and malty. The point here is that choosing a strain with the attenuation you want can get you a very distinct flavor. Calculating and estimating apparent attenuation is easy, given the example above:

% Attenuation = (original gravity - final gravity) / original gravity = (50 - 18) / 50 = 65%

Temperature

No shocker here, certain strains like certain temps to ferment optimally. This can be broken down into 2 categories: ales (62-70F) and lagers (48-59F). There are also hybrid strains that like mid ranges as well, such as kolsch yeast. As a general rule, you can avoid any potential off flavors in the beer from stressing the yeast by keeping the temperature of the fermentation at the low end of its range. The cooler you keep the yeast the slower it will ferment, but if you push the temp too high, the yeast will stress quite a bit.

Flocculation

Given in low, medium and high, flocculation is the ability of the yeast to clump up and fall to the bottom of the fermenter. Higher clumping will lead to clearer beer, but also may result in stalled fermentation by falling too quickly if not pitched at the proper amount. A low floccing yeast will leave a very cloudy beer typical of weizens and wheat beers.

Alcohol Tolerance

Most commercial yeast strains these days have very high tolerance to alcohol levels. It might be good to take a look at the rating before brewing a big beer, but most have a tolerance of at least 10% abv. There are imperial-type yeasts and champagne strains that would work well for high gravity beers.