There are three top leagues: Pro12, Top14 and the Premiership.
The Pro12 is made up of 4 Welsh regions, 4 Irish regions, 2 Scottish clubs and 2 Italian clubs.
The Top14 is made up of 14 French clubs.
The Premiership is made up of 12 English clubs.
There are also secondary leagues from these nations:
The English and French have the Championship and the ProD2 respectively. Teams go between these leagues and the Premiership/Top14 through promotion and relegation. I believe most the teams in these leagues are professional...
There are secondary Welsh/Irish/Scottish/Italian leagues down from the Pro12, such as the Welsh Premier Division or the Scottish Premiership. They cannot gain entry to the Pro12 and run separately from the Pro12.
There are numerous professional cups: the Heineken cup, the Amlin, the LV, the British and Irish cup.
The Heineken Cup is the big one. The top 6 English and French teams from their leagues gain entry. The two Scottish and Italian teams gain automatic entry. The three Welsh teams that finish highest in the Pro12 gain entry, same with the Irish. The winners of the Heineken Cup and the Amlin gain entry, but if they've already entered, then the place goes to another team from their country/league. (So when Leinster have won the Heineken or Amlin Cups, Connacht have gained entry so all four Irish teams enter the Heineken.) In total, that's 24. (6 English, 6 French, 3 Irish, 3 Welsh, 2 Scottish, 2 Italian, 2 more)
The Amlin Cup (or European Challenge cup) is made up of all the teams from the Premiership/Pro12/Top14 that didn't make the Heineken, plus teams from secondary leagues in Italy, Spain, Romania etc.. For the top teams, you're either in the Heineken Cup, or the Amlin. Since the pools are easy for some of the top teams in the competition, teams often focus their efforts on their league and then start to take it seriously when the Amlin reaches the knockouts.
The LV Cup (or Anglo-Welsh cup) is made up of the English and Welsh teams from the Premiership and Pro12. Since teams focus their efforts on the Heineken/Amlin or their league, this is usually treated as a way of testing your reserves and youth. If an English team from outside the top 6 of the Premiership wins the LV Cup, then they take the place of the 6th-placed English team in the Heineken Cup I believe.
The British and Irish cup involves the English Championship clubs, A teams from the Irish regions, the teams from the Welsh Premier Division and a few Scottish teams (not Glasgow/Edinburgh).