According to this [site][1] and the [Honor FAQ][2] 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)

![profile indicator][3]

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

![Great Teammate][4]

Same thing but for being an honorable opponent

![honorable opponent][5]

These also show up during the loading screen

![enter image description here][6]

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


![popup][7]

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 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. 


  [1]: http://www.surrenderat20.net/2012/10/psa-honor-rewards.html
  [2]: http://na.leagueoflegends.com/honor
  [3]: https://i.sstatic.net/wpjjI.jpg
  [4]: https://i.sstatic.net/dnSHj.png
  [5]: https://i.sstatic.net/WG2Mm.png
  [6]: https://i.sstatic.net/pfNTn.png
  [7]: https://i.sstatic.net/pNLtu.jpg