(Some of the cards shown above were contacts made under my old call NV7J)
In the US, DX most often refers to a foreign country no matter how far the distance. I live in Lacey Washington and the nearest foreign country is Canada, about 100 miles distant from my QTH. But is considered a DX country.
The ARRL (Amateur Radio Relay League) has some definitions created in support of various contests and awards. For example, the ARRL DXCC award recognizes a DX contact as a contact with a DX entity outside of your QTH entity. For example I live in the USA and even though Alaska and Hawaii are parts of the USA, they count as DX entities and you can get credit in contests and awards for working and confirming them. There is a list of entities credited for DXCC that can be downloaded from the ARRL.
When making contact with a DX station you may hear them on a frequency like 14.195. They may be transmitting and listening on that same frequency. But if it is a rare or hard to get DX station they, most of the time, will announce they are working SPLIT. This means they will be listening for callers either up or down from their frequency. It is up to the DX station to make this announcement. Listen for it before you transmit. And for heaven sakes do not scream into your microphone UP, Up, UP you dummy.
Please give the DX station a true signal report so they can determine how strong they are being heard in parts of the world. Listen, Listen and do not call over other. If the DX asks "Who was the 3 Oscar?" please do not add QRM by putting you call in there. Sometimes the DX station will ask for only calls from areas of the world like Japan only, or Europe. Also sometimes because of the number of hams calling from the US the DX may ask for stations by the numbers 6s only, for example. Those of us who have calls outside our real area like me W3OZ living in 7 land, can get an extra call ever so often. But if you call that way be aware you may not be found in the log as some DXers keep logs by call area so you may want to list you call as W3OZ/7 if you made you contact while DX was calling for 7s.
OK, so now you THINK you have made a contact with the station you were calling. If you are not working toward an achievement like DXCC or one of the many other awards, just put the details of the QSO in you log and be happy. But if you are working toward an award of some kind you are going to have to confirm it with the agency like the ARRL or in some cases CQ magazine and others.
First, maybe you thought the DX station came back to you but maybe you didn't hear it or the DX station didn't log you in the official log. If you are found not to be in the log and you are sure you had a good contact, sometimes writing the DX station or the manager if it has one with the exact detail can get you a good contact. I was a manager for an African station and found that sometimes the notation of the QSO was miss-logged by not hearing the station call sign correctly and if can be seen in the log that the operator heard the station but was off by one letter in the call sign you may get that coveted QSL card after all.
So how do you send for the DXs QSL card? Use resources on the internet like QRZ and DX bulletins to find the mailing address for the station in the foreign entity or manager if there is one. Put your QSL card in an envelope with the details about the QSO in a legible fashion. If the DX station or manager has thousands of cards to verify and can't read your card easily he or she may just toss it in the garbage. Put a return envelope with your address on it, and a green stamp or two. What is a green stamp? No, not an S&H green stamp but American money. Be generous as the DX station may have much high postage rates than we do here in the US. DO NOT put your call letters or anyway to identify you as a ham as thieves will look for things like that to tell them that there well may be money inside and the DX station will never see your card or money. You shouldn't have to do that for a manager in the US, but adding a little monetary thank you, is of course welcome. It costs thousands these days to do a DX expedition.
Keep and maintain a good log. Maybe a list of entries you need by band so you can check then off as you get confirmation by the organization you are applying to for an award. Do not think it is a done deal until confirmation is received. More than one time I thought I had worked a DX station to find out it was a pirate station. What is a pirate station? When some operator says he is some place in the world but really not there, or the operation is not authorized by the proper government authorities, to operate, this is called a pirate station and you will not get credit for it. I was not credited for a very rare entity at a DX convention by the ARRL as the card was given to me by the operator there and not mailed. I eventually did get credit by having the DX station write to the ARRL.
Now here is the last thing I want to bring up about DXing. One of the topics will not sit well with some of my audio friends. I have never had the pleasure to be on a DX expedition but have done some special stations and contests. I have talked to some folks who have been on expeditions and they tell me that in rare places when they call out, that the S meters on their rigs go to 60 over or worse and it is hard to pick out stations in the mass of sound. When talking to my friends each afternoon on 14.178 I want smooth easy listening audio. Yes like you hear on TV and broadcast radio. Don't give me any crap about we are communicators not broadcasters etc. We should be skilled enough to provide good quality audio when it is appropriate and what I call DX audio at other times. Also don't give me the crowded band shit either. I have been jammed by the 1930s style old fashioned Bell Labs ear splitting audio almost on a daily basis.
But it is harder to put a 4K signal with some lows in it to a far away DX station than a narrow signal. And, the DX stations are not used to hearing broad type audio in a pile up. I have tested this many times and I can tell you to put on your narrow microphone, put a narrow bandpass on your transmitter, put a lot of compression on, sound like hell but you will have a better chance of getting the DX you are chasing.