ristiuMMask made an awesome video teaching all about the honor system: https://www.youtube.com/watch?v=EIAeisuevwc
It's got so many good life lessons as well as covering nearly all the facets of the honor system, but if you don't want to watch all 24 minutes, here's the gist of ribbon part.
Basically, in order to get a ribbon, you must consistently receive honor, it must be high-value honor, and it must be the right kind of honor.
How do you receive honor? That part is the most tangible part. You should be receiving honor in a lot of your games if you want to be on the track towards getting a ribbon (and keeping one). If you get honored in enough of your games for a good amount of games, you should eventually get a ribbon. This means that you need to be a nice person. Talk to your teammates, congratulate them on plays, give friendly advice, fill roles, play well, be a good sport, don't insult people, calm down ragers, and don't express negative feelings in chat. If you want a ribbon, you need to show that you're a real gentleman or lady, and that takes hard work.
Now, you also need to be receiving high-value honor. The honor count means little. It is a simplification of your actual honor score. The honor that every person gives will be different in value. Firstly, strangers' honor is worth the most. This means people that aren't on your friend list nor people that are in your premade (whether or not they are friends). Secondly, the rate at which people honor (honor per game) determines how much their honor is worth. The lower one's honor per game, the higher one's honor is worth. If you get honor from someone who honors lots of people every few games, that honor will be worth somewhat less than the honor gained from someone who hasn't honored anyone in the last fifteen games. However, if they were both from strangers, both will be higher than the honor gained by you from your friend, even if your friend doesn't honor often. The time it takes for honor to reset in value is unknown.
Lastly, you need to be receiving the right type of honor. The Great Leader Ribbon requires Friendly, Helpful (less), and Teamwork honor. The Great Mentor Ribbon requires Friendly (less) and Helpful honor. The Great Teammate Ribbon requires mostly Teamwork honor. The Honorable Opponent Ribbon requires you to have a high amount of Honorable Opponent honor.
I hope this will help you become a more honorable member of the League of Legends community! And remember, achieving the ribbon is only the first step. Ribbons can decay if you are not consistently showing yourself as a honorable person, so...
stay positive!