Every rank has 5 tiers (like Bronze has Bronze 1 - 5, same with all the other ranks) The lowest tier in a rank is 5 (So for example Bronze 5 is the lowest you could ever be). As you win games, you get lp and then once you reach 100 lp you go into your promos. If you are in tier 2 - 5 for a rank (ex: Bronze 2 - Bronze 5) then the promos would only require you to win 2 out of 3 games, and if you do then you go up a tier. If you are at the tier 1 of that rank (Ex: Bronze 1) then you will have to play 5 games, and win 3 out of the 5 but if you do then you go up a rank (So Bronze to Silver).
If you fail your promos then you will be close to 100 of your current rank (Example, I recently failed my promos and was put at 90 lp) and when you re-enter promos for that rank again you will have 1 automatic win that counts towards your win 2/3 or 3/5.
Ranks go like...
Bronze 5 - Bronze 1
Silver 5 - Silver 1
Gold 5 - Gold 1
Plat 5 - Plat 1
Diamond 5 - Diamond 1
Masters
Challenger
I can't really explain much for Master/Challenger since they have a different system. I believe Challenger is just the top X amount of players (I don't know the amount of Challengers there are, 200?) I think Masters was added for the people who were past Diamond 1 but not high enough to rank for the top Challengers or something.