As someone that had similar questions a few years ago, I'll try and explain a little too.
--Each country has their own domestic league. Examples in Europe: Premier League in England, La Liga in Spain, Bundesliga in Germany, Serie A in Italy, etc. The US has MLS.
-- These leagues typically have lower tiered leagues (think like MLB for baseball being the top league and then your various minor leagues like AAA, AA, etc.)
-- Most of these leagues have 20 teams. Each team plays every other team twice during the year, once at home and once away. That's your 38 weeks. The season starts in August and goes through May. It's a very long season with only two months off.
-- There are promotions and relegations between all the leagues in a country. Thus, the three teams that finish at the bottom in the Premier League will be relegated down to the lower league next year. The three top teams in that lower league will get promoted into the Premier League. This is a very big deal and not something you see in American sports. Imagine the Atlanta Braves moving down to the Triple A league because of how poorly they do and the top Triple A team moving up to MLB next season to give you somewhat of an idea.
-- You have probably heard of some major clubs. Manchester United, for example, is in the Premier League. Real Madrid plays in Spain in La Liga. Bayern Munich plays in Germany in the Bundesliga. These teams only play teams outside of their domestic league during tournaments. The tournaments, like UEFA Champions League, happens yearly and runs during the same time as the season and runs for a long time. Winning the UEFA Champions League is prestigious.
--Each country typically has a yearly domestic tournament as well. In England, for example, that's the FA Cup. It's made up of all the club teams in all the divisions (i.e. MLB teams together with all the minor league teams). Thus, a club like Liverpool could be involved in their domestic league (Premier League), their domestic tournament (FA Cup) as well as the UEFA Champions League (or lesser Europa League) all at the same time. The greatest feat is winning all three in a single season ( a treble). Only one team each year has the possibility of doing so (UEFA Champions League winner).
-- Some of the players that make up these club teams also typically have International team duties. For example, Messi not only plays for Barcelona as his club team in Spain, but he plays for Argentina in international competitions and friendly matches because that's where he is from and where his citizenship is. Players can only play for one country throughout their career (for the most part).
-- Players can, and do, play for different clubs throughout their career. Cristiano Ronaldo (who is from Portugal), for example, played for Manchester United in the Premier League and then joined Real Madrid in La Liga where he currently is. Players can play for virtually any club team or league in the world. Some leagues are considered more prestigious than others.
-- Leicester winning is such a big deal because they were only recently promoted back into the Premier League and they were almost relegated last year. To now finish at the very top of a 38 week season ahead of giant clubs like Manchester City, Manchester United, Chelsea, Arsenal, Liverpool, etc. who have better players and far more money to spend is why this is such an incredible thing and why the odds of doing so are so low. Cleveland winning a Super Bowl would pale in comparison to this. To give you an idea, the Miracle on Ice was only 1000:1 underdog accomplishment compared to this at 5000:1.
Hopefully that helps and other veterans can correct anything I screwed up. Also, this thread is way cool and always welcoming to people that want to learn. The only time you see people not being nice is when it comes to club rivalries in here which is where most of the fun is. See Tottenham fans this year, for example.
Eta -- The scoring was explained above. That's why there are no playoffs. That said, determining the winner can come down to the wire and be decided in the last match of the season. There was an incredible finish just a few years ago, for example, with 2 late goals in the last game determining the winner --
https://youtu.be/81bv_gF4j5k