Five common mistakes new homebrewers make and how to avoid them!
**EDIT 10/14/2013 - Very cool! Some of the info that I provide below was used for this Popular Mechanics article: "4 Home-Brewing Mistakes Most Beginners Make", for which I originally wrote the material (I was not made aware that it was going to be used before the article was released, I thought my email just went into the ether!)
I've been homebrewing since 2009 and also organize and judge local homebrew competitions, often for the SF Homebrewers Guild, an organization I founded in Feb 2012. In addition, I'm a Certified Cicerone, which is a certification program for beer service similar to the advanced certifications a wine sommelier attempts to earn. I'm very familiar with the off-flavors that can result from poor brewing technique (they're on parade during any homebrew competition judging!). Part of a judge's job is to give the brewer feedback for how to avoid those errors in the future.
Here's a quick rundown of the most common homebrew mistakes which result in off flavors and "that homebrew flavor."
1) Not focusing enough on proper sanitation
Sanitation sanitation sanitation. The primary foes of all brewers are wild yeasts and bacteria. You can do everything else perfectly during your brew day, but if your sanitation practices are not up to par at every stage, you'll likely end up with a beer that's been fermented by something other than your brewer's yeast. The result is typically a bad beer that can be sour, thin bodied, fizzy (bottle gushers!), and can have phenolic flavors such as "plastic" or "band aid" or even other nastier ones (beer with a "fecal" aroma, anyone?). First of all, you need to understand that you can't sanitize brewing gear that isn't clean, therefore it's a two step process - properly cleaning and then sanitizing. Read this bit from John Palmer's How To Brew book for more details on cleaning and sanitizing and the various products that you can use. Also, remember to replace your plastic brewing gear each year - when plastic gets scratched it creates microscopic hiding places for wild yeast and bacteria and it becomes impossible to sanitize the equipment. It's cheap to replace plastic tubing and buckets, it's not worth risking the beer you put so many hours of work into! Something to keep in mind is that infected or otherwise bad beers can't harm you like home distilled liquor can, they just taste bad, so don't be afraid of tales about people going blind from "bath tub gin!"
2) Use a "rolling boil," do not leave a lid on your boil kettle, & remove your partial mash grain before raising the temp to boiling
Malted barley has precursor chemical in it that, when boiled, converts to Dimethyl Sulfides (DMS), a chemical that tastes like cooked corn and can ruin a beer. Luckily, DMS is very volatile and a good rolling 60 minute boil will drive most of it off - but if you leave the lid on your pot, it will all just condense on the lid and drip back in. Another boiling-related tip - if you're making a "partial mash" brew with extract and a little bag of grain, always remove the grain bag before raising the temp up to boiling. Also, don't squeeze the liquid out of that bag when you pull it out (just dunk it in and out a few times and let it drain). The reason you don't want to squeeze the bag or raise the temp above 170F with the grain in the water is that you'll leach tannins out of the grain husks and it will give your beer an astringent mouthfeel similar to how red wine or tea is sort of bitter and makes your mouth feel dry.
3) Ensure good yeast health
Brewer's don't make beer, yeast does! We just make the wort and manage the yeast after that. Ultimately, the brewer is a yeast wrangler. Use a good healthy pitch of fresh yeast for your beer. For new brewers, packets of dry yeast can be easier to work with, though liquid yeasts offer more variety. If you're making a high gravity beer you'll need more than a single pitch of yeast. For any brew, don't expect some old pitch of liquid yeast that has been in your fridge for three months to do the job, go with a fresh pitch. If you under-pitch or pitch unhealthy yeast, your yeast will be stressed during fermentation and will create off flavors in your beer. To give the yeast a little boost, toss some yeast nutrient (Servomyces) in your wort at the end of the boil. Also, make sure to aerate the wort once it is cooled down and in the fermentor, most brewers do this by rocking their fermentor back and forth for a few minutes after getting the cool wort into the fermentor. Yeast needs oxygen to divide and will absorb it all, this happens very soon after you pitch it in, before you see all the bubbling activity from primary fermentation. That said, don't let your beer have much contact with oxygen after primary fermentation begins or you risk papery-like oxidation off flavors. Here's some additional info, again from John Palmer's How To Brew - preparing yeast, yeast forms (dry & liquid), yeast nutritional & oxygen needs (read the following few pages).
4) Control your fermentation temperature
Next to proper sanitation, there's no question that fermentation temperature control is the most important variable. Yeast likes to work within a certain temperature range, the recipe you're using should give a specific preferred fermentation temperature. Typical American ale yeasts like a temperature of about 68-73F. Fermenting at higher-than-recommended temperatures will typically cause the yeast to create more esters and phenols, leading to fruity or spicy aromas and flavors that may not be appropriate for the beer you're making. Of even greater import is keeping the temperature even for the duration of the fermentation. Wrap your fermenter in a blanket and stick it in a dark closet in the center of your house or apartment to minimize the likelihood of temperature swings. If the temp drops during fermentation, the yeast will likely stop working and you'll be left with a sweet brew that hasn't been fully fermented or a beer with remaining off flavors that the yeast would have scrubbed out had they been given a few extra days to finish the job. You should be able to figure out if your fermentation did not complete by checking your Final Gravity with the hydrometer that came with your starter kit. A target FG should be provided in your recipe, if your FG is much higher than that, you may have an incomplete fermentation. Be careful about bottling such a beer, if the fermentation continues or starts back up in the bottle you could end up with "bottle bombs." There are lots of good reasons to take your gravity readings!
5) Your local homebrew shop and homebrew club are a resource for you
This isn't exactly a tip for avoiding a specific mistake, but it can keep you from making mistakes! The guys and gals at the local homebrew shop or club want to help you on your homebrewing journey. They have a wealth of knowledge they will happily share, don't be intimidated, ask them all your questions even if you think they'll reveal that you're new to brewing. You can even bring them your homebrew and they will provide feedback. The local shop can put you in touch with local homebrew clubs if you're not sure where to look. So don't worry about saving a buck ordering online, support the local shop and get to know folks in your homebrewing community if you really want to learn more and make great beer!
BONUS) For the new homebrewer, these are the best books to have on the shelf for brewing information and recipes.
Best source for general homebrewing knowledge: How to Brew by John Palmer (I realize it appears that the full book is online, I even posted links to it above, but the updated paper version has significantly more, and newer, information. Just buy it!)
Best source for tried and true classic recipes: Brewing Classic Styles by Jamil Zainashef & John Palmer
Best source for exciting, inspiring, and whacky recipes: Radical Brewing by Randy Mosher
There are loads of other books out there, but these are all you'll ever need...that is, unless you decide to go deeper down the homebrewing rabbit hole. It goes as deep as you'd like, trust me, one day maybe you'll be testing to become a BJCP judge! One of the beauties of homebrewing is that there's always more to learn, always ways to improve, and that's not to mention the awesome community of folks who are into homebrewing. It's a hobby that can take you as far as you'd like to go with it.