DX is radio slang for distance. At first, when everyone including amateurs used fairly long wavelengths, and amateurs struggled very hard to make contacts across the Atlantic Ocean (hardly dreaming then of the Pacific!) DX really meant distance. When amateurs began to use short wave, it was not long before many of them could make contacts halfway around the world, which seemed about as distant as one could go. Since then, a method by which radio amateurs assess their accomplishment in the world of DX is by the number of "countries" they have contacts with. Countries is in quotes, because for amateur radio purposes an ordinary political definition of a country is not acceptable. The rules defining an amateur radio "country" are too arcane to try to go into here. Suffice it to say that in addition to France, Germany and Japan, Hawaii and Alaska and the usually uninhabited Canadian island in the Atlantic Ocean called Sable Island are countries to the radio amateur. There are more than 300 of them (one is a house in Rome that is the headquarters of the Knights of St. John of Malta and another is the United Nations headquarters in New York City), and at times some amateurs have had contacts with them all! It is, unfortunately, not enough for an amateur to say he/she has had a contact with another amateur in a particular country; it is also necessary, at the very least, to be able to show a card, called a QSL card, from the other amateur attesting to the conditions of the contact (in a few cases a dispute may even arise as to whether the other amateur was properly licensed to operate in the place he/she said they were in). I accumulated such cards from more than 150 countries before my enthusiasm for this particular aspect of ham radio declined. It has a lot in common with stamp collecting, which is an OK hobby too!
In addition to the kinds of amateur operation I have described here and elsewhere, there is "contesting". Contests come in all sorts, but almost all are to see who can have the most contacts with other stations under specified conditions. Contests are tests not only of technical knowledge and equipment but also of operating skills and particularly of endurance. Contests are not popular with those who don't participate in them, because some of the contests cause interference with other amateur operations, but they don't go on all the time, so the contestants and non-contestants can both recover from the ordeals.