First Wort Hopping
An old yet recently rediscovered process (at least among homebrewers), first wort hopping (FWH) consists of adding a large portion of the finishing hops to the boil kettle as the wort is received from the lauter tun. As the boil tun fills with wort (which may take a half hour or longer), the hops steep in the hot wort and release their volatile oils and resins. The aromatic oils are normally insoluble and tend to evaporate to a large degree during the boil. By letting the hops steep in the wort prior to the boil, the oils have more time to oxidize to more soluble compounds and a greater percentage are retained during the boil.
Only low alpha finishing hops should be used for FWH, and the amount should be no less than 30% of the total amount of hops used in the boil. This FWH addition therefore should be taken from the hops intended for finishing additions. Because more hops are in the wort longer during the boil, the total bitterness of the beer in increased but not by a substantial amount due to being low in alpha acid. In fact, one study among professional brewers determined that the use of FWH resulted in a more refined hop aroma, a more uniform bitterness (i.e. no harsh tones), and a more harmonious beer overall compared to an identical beer produced without FWH.
Bittering
The primary use of hops is for bittering. Bittering hops additions are boiled for 45-90 minutes to isomerize the alpha acids; the most common interval being one hour. There is some improvement in the isomerization between 45 and 90 minutes (about 5%), but only a small improvement at longer times ( <1%). The aromatic oils of the hops used in the bittering addition(s) tend to boil away, leaving little hop flavor and no aroma. Because of this, high alpha varieties (which commonly have poor aroma characteristics) can be used to provide the bulk of the bitterness without hurting the taste of the beer. If you consider the cost of bittering a beer in terms of the amount of alpha acid per unit weight of hop used, it is more economical to use a half ounce of a high alpha hop rather than 1 or 2 ounces of a low alpha hop. You can save your more expensive (or scarce) aroma hops for flavoring and finishing.
Flavoring
By adding the hops midway through the boil, a compromise between isomerization of the alpha acids and evaporation of the aromatics is achieved yielding characteristic flavors. These flavoring hop additions are added 40-20 minutes before the end of the boil, with the most common time being 30 minutes. Any hop variety may be used. Usually the lower alpha varieties are chosen, although some high alpha varieties such as Columbus and Challenger have pleasant flavors and are commonly used. Often small amounts (1/4-1/2 oz) of several varieties will be combined at this stage to create a more complex character.
Finishing
When hops are added during the final minutes of the boil, less of the aromatic oils are lost to evaporation and more hop aroma is retained. One or more varieties of hop may be used, in amounts varying from 1/4 - 4 oz, depending on the character desired. A total of 1-2 oz. is typical. Finishing hop additions are typically 15 minutes or less before the end of the boil, or are added "at knockout" (when the heat is turned off) and allowed to steep ten minutes before the wort is cooled. In some setups, a "hopback" is used - the hot wort is run through a small chamber full of fresh hops before the wort enters a heat exchanger or chiller.
A word of caution when adding hops at knockout or using a hopback - depending on several factors, e.g. amount, variety, freshness, etc., the beer may take on a grassy taste due to tannins and other compounds which are usually neutralized by the boil. If short boil times are not yielding the desired hop aroma or a grassy flavor is evident, then I would suggest using FWH or Dry Hopping.
Dry Hopping
Hops can also be added to the fermenter for increased hop aroma in the final beer. This is called "dry hopping" and is best done late in the fermentation cycle. If the hops are added to the fermenter while it is still actively bubbling, then a lot of the hop aroma will be carried away by the carbon dioxide. It is better to add the hops (usually about a half ounce per 5 gallons) after bubbling has slowed or stopped and the beer is going through the conditioning phase prior to bottling. The best way to utilize dry hopping is to put the hops in a secondary fermenter, after the beer has been racked away from the trub and can sit a couple of weeks before bottling, allowing the volatile oils to diffuse into the beer. Many homebrewers put the hops in a nylon mesh bag - a Hop Bag, to facilitate removing the hops before bottling. Dry hopping is appropriate for many pale ale and lager styles.