Ranked system overview, modern changes
League of Legends has featured a large number of changes to the ranked system since the other answers have been posted. Notably:
- The ELO system is gone.
- The Grandmaster, Emerald, and Iron ranks have been added. This changes the ranks to be ordered as:
- Challenger, Grandmaster, Master, Diamond, Emerald, Platinum, Gold, Silver, Bronze, Iron
- The fifth divisions of each rank have been removed.
- Promos are gone. When you exceed 100 LP in a given division, you move up to the next-highest division. If this causes you to change ranks (e.g. you exceed 100 LP in Gold I) you will promote to the next rank immediately (in this case, Platinum IV)
- Placements are wildly different. Each rank queue places you over the course of your first 5 games in that queue (per reset). You earn accelerated LP while placing, and you cannot lose LP. This tops out at Emerald I, and cannot exceed it.
On the topic of decay...
Decay has also received changes. I'm sourcing the following answers I'll give from the support article on Placements, Promotions, Series, Demotions, and Decay from Riot Games' website.
At what Elo does your Elo have the possibility to decay?
Addressing this question first, decay only applies to the Apex tiers (Challenger, Grandmaster, Master) and Diamond. If you're in Emerald or lower, LP decay does not occur.
How long (to the day) without playing a Ranked game does it take for your Elo to decay?
If you play a ranked 5v5 does it reset the Elo Decay timer for your solo queue?
Rank decay works on a "banked days" system. Each ranked match you play in a rank that features decay, you bank a certain number of days in accordance with your rank. These days are consumed each day you do not play a ranked game. After you've reduced your banked days to 0, you will experience LP decay for every day thereafter that you do not play a ranked match.
If you're in Diamond, you can bank up to 28 days (banking 7 days per match). After those 28 days, without playing another match, you will experience rank decay.
In the apex rank tiers (every rank above Diamond), you can only bank up to 14 days (you generate 1 day for every match you play). Afterwards, you will experience decay.
At what percentage does it decay?
No percentage, just a flat amount. For Diamond, this is 50 LP per decay day. In apex tiers, it's 75 LP. If this causes you to go below 0 (or fall out of the thresholds for Apex tier ranks), then you demote immediately. Demoting from an apex tier rank to Diamond via decay is special-cased, and causes you to drop to Diamond II immediately, rather than Diamond I.