Some hams just enjoy working DX. Some combine it with collecting QSL cards and applying for DX awards (some refer to them as "paper chasers"; I think that's a derogatory term; regardless... there's room for everyone in this hobby).
I'll cover some of the common awards and some of them that are not so well known.
The ARRL's DX Century Club (DXCC) is the "gold standard" for the DXer . If you go to a DX banquet at Dayton or Visalia, there will be an exercise where everyone with 100 or more countries worked will stand up... remain standing if you've worked more than 125... 150... 175... 200... etc... 300... 310... 320... 330... etc... 340... 350... 360... 370... up to 402. While there are 340 countries on the "current list", there have been another 62 countries that have been removed from the list (referred to as "deleted countries"). Oddly enough, there are "old-timers" out there who have "worked 'em all."
The mixed (basic) award is earned by confirming 100 DXCC countries regardless of band or mode. In all, there are 18 awards in this program; plus 5BDXCC, Honor Roll, and #1 Honor Roll.
I presently have the DXCC award for Mixed, CW, Phone, Digital, 160, 80, 40, 30, 20, 17, 15, 12, and 10 meters (9BDXCC; just a few left for 6m). I have the 5 Band DXCC award (5BDXCC) with endorsements for 3 additional bands for a total of 9 bands. My challenge total is 2002 confirmed "band countries." Including deleted countries, I've confirmed the 343 countries I've worked. I reached the DXCC Honor Roll in 2015. I've got 5 countries left for "#1 Honor Roll." (They say, "If you live long enough, you'll work 'em all") I've been active in this award since 1974 and I enjoy it very much. It's been the key thing that has put excitement into ham radio for me.