Oh yeah, well that's actually not so difficult
Once upon a time, there was a big darts family, the BDO (British Darts Organization). Then there was a fight about money and stuff (what else is new), and a handful of players (including now legend Phil Taylor) broke off and formed the PDC (Professional Darts Corporation). For quite a while, the PDC was more obscure and, apart from Taylor, the BDO had the best players. This, however, changed, and more and more players switched to the PDC simply because there's a whole lot more money to be made there.
Especially since 10-15 years there has been a mass exodus from the BDO to the PDC, resulting in now basically all of the world's best players playing in the PDC, their tournaments being shown on TV and the BDO being the obscure one. BDO has the Lakeside (formerly known as The Embassy) as it's world championship, the PDC has the, well, World Championship as it's world championship.
The level in the BDO is way lower nowadays and maybe 1-2 of their players would have a shot at being a top-20 player in the PDC. Also, whenever someone in the BDO is playing well, the rumours about him switching to the PDC start and in some cases turn out to be true. The advantage of the BDO is seeing players from more countries, the big disadvantage is that quite often these players bring a level that, for someone who is used to the PDC, is almost painful to watch (imagine watching Diamond League athletics all the time and then watching national events with 100m being run in barely under 11 seconds by the men).
WDF is the darts federation that's part of SportAccord (or whatever it's current name is, GAISF?), but doesn't have their 'own players' like the BDO and PDC. Mostly it's BDO players in their tournaments because PDC players are busy with serious stuff (like the World Grand Prix of this week), but I do think PDC players are welcome as well (two Russians who played in the WDF World Cup this week are PDC players).
So, in very short:
- PDC and BDO are arch enemies
- PDC is the top level, this is where the action's at
- BDO has a way lower level, but does have some more attention to women's darts (PDC has no women's tournaments) and to youth development (PDC does have a Youth Tour, but only for the best of the best, BDO is more about 'spreading the sport')
- WDF is the body that, if darts would ever be Olympics, would probably represent them at SportAccord/GAISF/IOC etc
Fun fact:
Every year there is the Grand Slam of Darts. The best PDC players in their ranking + finalists and winners of majors play against 8 BDO players (finalists/winners of majors and so on), which leads to some always interesting direct PDC-BDO matchups, outside of this event rather rare/non-existent.