Only EU and NA has LCS, LCK(Korea) and LPL(China) etc. have different formats.
Each split has a promotion tournament before it starts, which is the way that teams change in the LCS. From the bottom three(summer) or four(spring) teams of the split, the worst team automatically drops out and can't participate in the promotion tournament. The others have to play against Challenger Series teams to keep their spot. If they lose, they will be dropped from the LCS and the Challenger teams will take their place. Challenger Series is for teams that are on top of their regions respective ladders.
All 6 of the top teams of a split go to the regional playoffs at the end of each split. For both playoffs you're rewarded Championship points. The team with the most points in the end of the season gets to go to worlds. The team who wins summer playoffs gets to go to worlds. Then there's 4 teams left, who have to play against each other in Regional finals, which is a bo5 single elimination tournament with placings based on their championship points. So, let's say we have Teams 1, 2, 3 and 4. 1 has the most points, 4 has the least. So the bracket would go like this:
Team 3 vs Team 4 -> Team 2 vs Winner -> Team 1 vs Winner
Worlds has 16 teams who have to qualify to play there. LCS qualifying I just explained, other regions qualifiers I'm not so familiar with. This year the tournament has places for 3 teams from Europe, 3 from NA, 3 from China, 3 from Korea, 2 from Taiwan and 2 for International Wildcards, aka the smaller regions(Brazil, Japan, SEA, Latin America, Turkey, Oceania and CIS). Before the bo5 single elimination bracket stage which decides the winner of the tournament, there's a group stage, where the teams are divided into 4 groups of 4 and play against each other in round robin format. In groups each team plays against the others in its group twice.
edit: There's no teams on this yet, as qualifiers are still going on, but it explains the structure of the actual tournament pretty well.
http://lol.esportspedia.com/wiki/2015_worlds#Results