American Pale Ales are best known for their crisp citrusy hop notes and balancing malt character. This style seems to have emerged from the strong hop-producing regions at the beginning of the American craft beer movement. It quickly exploded to be one of the most popular styles in the United States, and you can find at least one example from almost every brewery in the country.
While there are many regional versions and characteristics, the underlying theory behind this brew is unified: a strong pale malt bill with just a bit of caramel or bready character, a mild American yeast strain and a strong (but not overwhelming) American hop profile.
The hops need to be the centerpiece of this style, but it's important not to let the flavors get unbalanced. The major style guidelines put the bitterness of an APA at no more than 45 IBUs (no less than 30). I tend to enjoy a hop-heavy pale ale, so I usually brew them to be about 45 IBUs. The variety you use should have predominate citrus or pine character. The classic American varieties include Cascade, Columbus and Centennial, affectionately known as the "three Cs". If you're looking for more piney characteristics, you will find it in Simcoe or Chinook hops. For the braver brewer, more extreme hops like Amarillo or Citra are an option.
Equally as important as the varieties of hops you chose is the way you use them. A large portion of the hops should be used late in the boil. The later the hops are added to the boil, the more impact they have on flavor and aroma. You can actually add hops up to the point where you turn off the burner, called "flame-out", and even while the wort is cooling if you are very careful about sanitation.
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Hops added at the beginning of the boil contribute more to the apparent bitterness of the beer and less to the flavor. Since balance is important when brewing an American Pale Ale, I like to create a hop schedule that has an approximately equal portion of hops added 3 or 4 times throughout the boil. Here's an American Pale Ale recipe for the beginner level homebrewer.
Keep in mind, the best path to success is to practice proper sanitation throughout your brew day. While the wort is boiling, fill your sanitation bucket with clean tap water and mix your sanitizer to the right dilution following the directions on the bottle. Anything that touches the wort after it begins cooling should be cleaned and then soaked in the sanitizing bucket for at least a couple minutes. This includes your airlock and stopper, as well as any spoon, thermometer or other utensil you might want to use. Transfer a portion of the sanitizer to your clean fermentation vessel—either a fermentation bucket or carboy. Swirl the sanitizer around in the fermentor so that it touches every surface, wait a few minutes, and do it again. If you're using a bucket for fermentation, be sure to get the lid as well. Transfer the sanitizer back to the sanitizing bucket and close up your fermentor. Don't open it again until you're done brewing and you transfer your wort into it.