Dry Hopping

Definition

Dry hopping is the process of adding hops into your beer after the onset of fermentation - either in the primary or secondry fermenter (if one is used). Because there is no boil, not bitterness will be imparted to the brew. However, the volatile oils that make up the aroma aspect of hops will be released into the beer. Usual amounts of hops are anywhere in the 1 to 2 oz range ... but if you like hops, knock yourself out!

Primary or Secondary

Dry hops can be added to your beer during either the primary or secondary stages of fermentation.
The advantage to adding during primary fermentation is that, if you plan on raking your beer to a secondary fermenter, you have the chance to take the beer off of the hops at exactly the time you would like. You also have the advantage of clearing the hops from the beer when you rack and again when you bottle.
If you choose to wait until fermentation has slowed, you have some advantages as well. Have noticed just how much agitation there is of the beer during your primary fermentation? Well, if you add hops during this time, it is conceivable that some of the volatile oils that you are trying to impart into your beer can be driven off by this physical action.
If you keg your beer, you can also add hops into the keg with the use of a sterile bag. I have read where people had a hook welded onto the bottom of the keg lid, and have hung the hop bag from that with good result. This makes it easy to remove the hops any time you like.
Weighing the benefit of mechanical filtration in the primary versus the chance of loss of some of those volatile oils, I would choose the secondary.

Is it Sanitary?

We work hard to keep everything sterile in, on, around, and near our fermenting beer. It may bring you pause to toss in some processed, wild plant cones directly into your hard-earned sterile environment. Never fear, however, because the chemical makeup of hops does not provide a friendly environment for a vast majority of bacteria and other nasties that you do not want in your beer. In addition, if you your dry-hop into the secondary, you're going to be dry hopping into an environment that contains alcohol - a slight insurance policy against bugs and critters. If you choose to dry hop in the primary, those wild yeasts are going to have a hard time fighting the billions of strong, already working, good yeast cells in there. All in all, I would say this is close enough to sterile.
One note - if you choose to put your hops in some kind of bag, do sterilize the bag completely!

What Hops to Use

Use any hops that you would use in the beer during the last 15 minutes of the boil. This is usually a low alpha aroma type hop (Cascade, Saaz, Fuggle, Hallertau, etc.). However, I have dry hopped with Centennial and Chinook as well. Try it out, there's no hard and fast rules here.
As for the type of hops you choose, it's really up to you - all will work. Hop plugs were specifically designed for dry hopping and each plug weighs 1/2 oz. for ease of measurement. Plugs and leaf hops will float. Pellet hops are easy to get into a carboy, and sink.
Note, when adding dry hops (especially pellets) you may get some foam as the CO2 nucleates on the hops and floats out of suspension. So, have some headspace available.