Per "The Mad Fermentationist"
1. Using the sanitizer that comes with a beer kit. This powdered sanitizer is slow and not especially effective. Instead get a no-rinse sanitizer like Star-San or Iodophor, which are faster and easier to use. Sanitize everything that touches your beer post-boil, and make sure it is carefully cleaned after each use (sanitizers are most effective on scrupulously-clean scratch-free surfaces). Keeping wild microbes out of your beer is the single most important step to brewing solid beer.
2. Starting with a recipe that is strong or unusual. Brewing a big complex beer is lots of fun, but play it safe on your first batch and brew something simple. High alcohol beers require more yeast and time. Interesting adjuncts add complexity to the recipe and process. These are things you don’t want to deal with on your first batch, so keep it easy.
3. Brewing with unfiltered, chlorine-containing tap water. If you are on a municipal water supply odds are that it contains either chlorine or chloramines. To remove them you can either charcoal filter or treat your water with metabisulfite, or alternatively use bottled water. One of the most common off-flavors I taste at homebrew club meetings is medicinal chlorophenol, which is formed by the combination of chlorine in the water or sanitizer and phenols from malt and yeast.
4. Squeezing the grain bag after steeping. Steep your grains in a small amount of water (no more than three quarts per pound) and then rinse them by either pouring hot water over the grain bag or dipping the grain bag into a second pot of hot water.Edit: I've had a couple people dispute squeezing being an issue in the comments. I've tasted some tannin-y beer from new homebrewers, but maybe it was just from a high water to grain steeping ratio.
5. Using liquid yeast. "Pitchable" liquid yeast cultures barely have enough cells to ferment a standard gravity beer on the day they are packaged, and their cells die quickly from there. A high quality 11.5 g package of dried yeast starts with as much as twice the cells as a fresh package of yeast from either Wyeast or White Labs, and retains high cell viability for much longer. While Fermentis, for example,claims a minimum of 6 billion cells per gram at packaging, the actual number tends to be much higher. Liquid yeast can produce great beers, but require a starter unless you are getting extremely fresh yeast and brewing a low-alcohol beer.
6. Not aerating the wort adequately. It takes several minutes of shaking for the chilled wort to absorb the ideal amount of oxygen to allow the yeast to complete a healthy growth phase. The healthier your yeast cells are the cleaner and quicker they will complete the fermentation.
7. Pitching when the side of the pot or fermentor feels “cool enough.” Use a sanitized thermometer to check the actual temperature of the wort before you add the yeast. Pitching when the wort is above 100 F is rare, but will kill the yeast. Ideally the temperature should be at or below your target fermentation temperature to allow the temperature to rise as the yeast grows and ferments. You can pre-chill the sanitized water you use to top-off after the boil to help bring the temperature down.
8. Fermenting at too high of a temperature. Take note of the ambient temperature of the room the beer is fermenting in, but realize that at the peak of fermentation the yeast can raise the temperature of the beer by as much as 7 F. Fermenting too warm can cause the yeast to produce higher alcohols and excessive fruity flavors. Letting the ambient temperature rise towards the high end of the yeast's range as fermentation slows helps to ensure a clean well attenuated beer, but for most strains is unnecessary. If you are unable to control the fermentation temperature, then choose a yeast strain that fits the conditions.
9. Racking to secondary. I know the instructions included in most kits call for transferring the beer from the primary fermentor to a secondary before bottling, but all this step accomplishes is introducing more risk of oxidation and wild yeast contamination. There is no risk of off flavors from autolysis (yeast death) at the homebrew scale in less than a month. At a commercial level the pressure and heat exerted on the yeast can cause problems quickly, but those conditions do not exist in a carboy or bucket.
10. Relying on bubbles in the airlock to judge when fermentation is complete. Wait until fermentation has appeared finished for a couple of days before pulling a sample of wort to test the final gravity. There is no rush to bottle, and doing so before the final gravity is reached results in extra carbonation. Once fermentation is complete and the beer tastes good, you can move the fermentor somewhere cool to encourage the yeast to settle out for clearer beer in the bottle.
11. Adding the entire five ounce package of priming sugar. In almost all cases this amount of sugar will over-carbonate the beer. Even for five gallons of beer this will produce too much carbonation for most styles and most brewers will end up with less than five gallons in the bottling bucket. Instead use apriming sugar calculator to tailor the weight of sugar you add to the actual volume of beer, the style of beer you are brewing, and the fermentation temperature.