As of patch 7.13, the Honor system has been updated! It now has a significantly different purpose, as well as some worthwhile rewards.
There are 6 levels of Honor (levels 0 to 5). Everyone automatically starts at Honor level 2. Whenever you gain Honor points (detailed below), you will make progress towards the next level of Honor. At level 2, you'll receive a few key fragments every couple days. Once you level up, key fragments become more frequent. In addition, you can start earning "capsules", which contain not only key fragments but also champion and skin shards. You even have a chance at getting the now Honor-exclusive Medieval Twitch and Grey Warwick skins from these capsules. Additionally, there will be end of season rewards for people who reached the highest level of Honor.
If you are ever punished (chat restriction, ban, etc.), your Honor level will go down to level 0. Once you're at Honor level 0, you will receive no rewards at all (no key fragments). If you play nice and earn your Honor back, at level 1 you'll start earning a handful of rewards (some key fragments, but still significantly less than Honor 2). If you end the season at level 0 or 1, you will be disqualified from getting any ranked rewards such as loading screen borders or Victorious skins.
So how do you earn Honor points? Anytime you play a matchmade game (which is pretty much every game mode except custom games), you will be shown an honor screen immediately after the conclusion of the game. On this screen, you can choose one person on your team to commend in one of three categories: "Tilt-proof" (stayed calm and friendly even in tense moments), "Great Shotcalling" (did a good job helping the team maintain focus on objectives), and "GG <3" (an all-around fun player to play with). Whenever you earn a commendation from another teammate, you gain a small amount of Honor points. But don't worry if you don't get many commendations - you will still earn small amounts of passive Honor exp for every game you play, regardless of commendations. Additionally, if everyone on the team offered commendations, everyone on your team will also get an additional boost of points.
You can read more details about the new Honor system on the wiki.