With so many cups in soccer, it's understandable that the casual fan will get confused and wonder which one is more important. So I'll try and help you out...
INTERNATIONAL SOCCER
1. World Cup (open to teams from all over the world)
2. Euro (open to European teams)
3. Copa America (open to South American teams)
There are others, but these are the big three that you'd want to pay attention to. Actually, I'd even go as far as to say that you should only really pay attention to the top 2... with Copa America, you're really going out of your way to find out info and maybe even catch some games so you're now becoming more than just a casual fan.
CLUB SOCCER
1. UEFA Champions League
This is Europe's version of the "World Cup" if you will. Only the best of the best can compete for this trophy. It's open to most of the champions from each countries domestic league, and depending how talented a country is, you can get more than one entry. For example, in Portugal the top two teams receive an automatic birth into Champions League with the third place team earning a bye into the final qualifying stage for Champions League.
2. UEFA Cup
While the best compete for the Champions League, the "best of the rest" will compete for the UEFA Cup. What's interesting about this is that in Champions League, there are four teams per group in the opening round. The two teams in each group that fail to advance to the next round in Champions League automatically drop down to the UEFA Cup to compete there. So again, pretty good quality of soccer, just not as prestigious as Champions League.
3. FA Cup
While every country has a domestic league and a domestic championship tournament... unarguably, England's FA Cup is the one that gets the most attention has the most prestige attached to it. Only teams registered to play in England can play for this trophy, in other words about 700 teams play in order to win this trophy. As such, many consider it a bigger honour to win the FA Cup than it is to win the English Premier League Championship.
That's basically the best of the best right there...