What's honor for?

According to this site and the Honor FAQ which I got most of my information from.

There are four types of honor:

• Helpful - There are those summoners who have the know-how to help you step up your game and are always willing to share. If one of your teammates makes you feel like you’re attending a master class on League of Legends, be sure to recognize him for being a helpful teammate.

• Friendly - Ever meet one of those summoners who – win or lose – is just a pleasure to play with? This acknowledgement goes out to those unsung heroes of the Fields of Justice whose friendly demeanor keeps everyone having a good time even when the chips are down.

• Teamwork - Some players really put the needs of the team over their personal interests. These players lead the way on the Fields of Justice, support struggling teammates, and are always the first to step up and select a role the team needs. This acknowledgement is for all those players who understand that a chain is only as strong as its weakest link.

• Honorable Opponent - Sometimes you run into an opponent who is humble in victory and graceful in defeat. Be sure to acknowledge those adversaries who embody the spirit of sportsmanship as Honorable Opponents.

You currently get some profile indicators, as well as other bonuses.

The profile indicator: (Icons left to right are: Friendly , Helpful , Teamwork , Honorable Opponent)

First bonus for getting that you are a great teammate is this during champion select:

Same thing but for being an honorable opponent

These also show up during the loading screen

The popup once you are alerted that you have received one of these alerts

Right now the exact amount of times you need to be honored is unknown. This is more then likely due to the honor votes being weighted in the background. Someone who honors just their friends votes count less then random people you play in matchmaking.

Right now the only way to loose honor is if you are reported to the Tribunal and found guilty of of a infraction. This resets all your honor to zero.

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.