DXing is making long distance contacts. The name of the hobby comes from DX, telegraphic shorthand for "distance" or "distant".If you are a 2 meter operator DX to you might be as near as a few hundred miles (or less) while and HF operator when chasing DX is looking for countries outside of his or her own. Generally though DXing is considered an HF activity.
To work DX successfully is help to listen for stations calling CQ. Pay attention to the style of operation used by the DX station and operate accordingly. If he is simply making one contact after another then don't try to engage him in a conversation when it is your turn. That will just cause frustration for everyone else. Using a DX cluster can also help you identify DX stations that you need. A DX cluster is an Internet site or packet station that reports current DX activity on the bands. DXpeditions are also good sources for new countries but be prepared to spend time getting through the pileups of other stations also attempting to call the DX station.
As you make DX contacts you will likely want to collect QSL cards. If you haven't already done so see the section on QSLing. As you collect QSLs from 100 or more countries you should consider the ARRL DXCC award. Check out the American Radio Relay League (ARRL) web site for more information.
DXers are always on the lookout to collect one more of the 338(currently) recognized individual DX entities. There are DX awards to compete for such as the ARRL DXCC(100 entities), and the DX Honor Roll(total number of entities).
Here are some sites for more information.