It’s important to bear in mind that international games frequently occur during the regular league (club football) season–although the international tournaments, like the Euros and the World Cups, take place during the summer (when club football is on summer break in Europe), games from the qualifying rounds of those tournaments are often held during the club season. Same thing for friendlies (i.e., “exhibition” matches).
So, every once in a while, the top leagues will be interrupted for a weekend of international fixtures, which usually happens just when your club is finally hitting its stride, and thus the break upsets your momentum and often leaves you with injured players who over-exerted themselves during a meaningless friendly.
One other thing to point out is that the players you see in the international teams typically play for the top divisions of a certain nation’s league (not necessarily their native country’s league), but you also have a rich assortment of lower divisions, the teams from which can often be promoted to a higher division (or relegated to a lower division). This system of promotion/relegation is unlike anything you’ll find in American sports, including the MLS, and is one of the features I find most engaging about world soccer.
How the system works, however, varies from one league to the next. Latin American leagues tend to have a rather Byzantine system which I can’t quite follow; in European leagues, it’s usually more straightforward–top teams winning promotion, bottom teams getting relegated–although nowadays you do have a “wild card” situation, as seen in the English League playoff system (as explained by GorillaMan).
The Triple is also called the Treble, which seems to be the preferred term in England.
And, technically, there is one team that has won a quadruple (if you limit the trophy count to major competitions, and disregard things like the World Club Cup as the Mickey Mouse cups that they are)–Celtic, back in '67. (admittedly, one can argue that the standards of Scottish football aren’t really as high as what you’ll find in England, or Italy, Spain, Germany… etc.)
And if you open it up to women’s football, the Arsenal Ladies have accomplished a similar feat. If only their male counterparts could do the same! (not in the foreseeable future, sadly).