How do Master and Challenger tiers work?

Master and Challenger Tiers work a bit different from the others (Bronze-Diamond).

First of all there is only 1 division in both but there is no upper limit in LP. Also the promotion from Master to Challenger happens automatically. Every 24 hours the MMR is checked and as soon as someone in Master has a higher MMR than a certain Challenger player he will take his spot.

In Master there is no player limitation but the Challenger tier is limited to 200 players.

Also in both of these special tiers LP decay happens a lot faster than in Platinum/Diamond. In Plat/Dia LP decay starts after 28 days of inactivity and drains LP every 7 days after while in Master/Challenger LP decay starts after only 10 days and drains LP every day after.

You can read more in the Riotgames Ranked-Play FAQ

Reaching master or even challenger proves that you are amongst the absolute elite in your current region. The skill difference between Challenger and Diamond 2 is usually bigger than the difference between Bronze 5 and Diamond 2 because it requires you to master absolutely every aspect of the game (or your main champion).

Master and Challenger are single division. There are 200 players in Challenger, while there can be an infinite number of players in Master (currently 600+ on EUW).

Once you reach Diamond 1 100lp, you have to win a bo5 to get into Master. You start there with 0LP, and earn points for each win, just like anywhere else. However, there is no stopping at 100 (top players in EUW have ~400lp). LP are not limited in Challenger either, with top players currently behing ~1200LP on EUW.

Every 24h hours, the top Master players are promoted to Challenger, while the lowest Challenger players are demoted to Master. The amount promoted players depends on their MMR compared to the ones of the lowest Challenger players.