Here's a few techniques to get back in their good graces. However, it is important to remember that favoring someone toward respecting you once again is a deep profound psychological and emotional process. This process can occur in a matter of moments or a matter of years, and sometimes may not occur at all.
Decide what you want. First of all determine whether or not you want to salvage this relationship, and why. "Moving on" is not always the only or the best choice; don't believe either that if you lose one real good friend, you'll never find another. Any relationship can be repaired in time, however if you are not fully ready, or if it's an unhealthy relationship that requires you to compromise your values or your self-respect, then now may not be the time. Realizing that the friendship is not at the stage you'd like it to be in can be a positive step toward eventually solving the issues between you and the other individual. Your self-esteem improves as you help another person gain a sense of belonging, and a feeling of mutual respect; bad friendships undermine your security and self-worth. A little dignity can go a long way into the early stages of repairing your rift. Don't fall into the trap of focusing on the negative qualities of the person they have had fallen out of favour with to justify your moving on; if you're going to let go of the relationship, devaluing the other person is just a cheap trick and not responsible.
Forgive as best as you can. When someone slights you, offends you, or hurts you deeply it is easy to want to respond in kind, retaliate, withdraw, become overtly defensive, sink into a depression, self-sabotage, or just feel plain mucky. The problem with these attitudes is that they are inaccurate reactions we use to "protect" ourselves from danger, real or imagined, but ultimately are in direct opposition to what we truly desire and are seeking: it either prevents us from communicating, or makes us behave defensively and hostile if we do, dooming our attempt to failure or a forced reconciliation that will not work in the long run. Ultimately conflict only breeds more conflict, and we must avoid this vicious cycle and free ourselves from a desire to hurt back, or to slump into quiet desperation. Ultimately we are trying to tip the balance of power in our favour, which in itself isn't necessarily a bad idea, except it neglects the humanity of the other individual involved in the dispute. Set your pride aside and bear in mind that the other person's perspective may be completely different - they may not even have realized that they hurt you. You may arrive at a point where you have forgiven as much as you can without communicating.
Make meaningful contact. Once you have decided that it might be a wise idea to re-establish contact, you need to write a note, call, or visit the person to convey with the utmost sincerity and honesty one primary message: "Our friendship is valuable to me, and I miss seeing you. Is there any way we can resolve what stands between us?" The point is simple - to convey your affection, express your willingness for reconciliation and invite them to an open, honest discussion. There is no need right now to air your grievances or even make elaborate apologies.
Find out what went wrong. You two must find a way to accurately understand the situation, with a logical, truthful, and rational perspective. Admit yourself that people are neither inherently evil nor inherently good (instead the harsh reality is that we lie somewhere in between.) This does not mean that we don't have innate human destructive tendencies... but it also does not mean that we don't have innate human productive tendencies either and presents with a more realistic view of reality. We're not always aware of the consequences or significance of our actions, either.
Understand the differences.Throughout history there have been plenty of examples showing both sides of the story; showing that differences can be accepted and understood. It isn't the differences themselves that are the problem, but how we deal with these differences. People even have different ways of handling conflict, and this once again needs to be understood before reconciliation can come about.
Take responsibility. You need to own up to your side of the story, since it always takes two to fight. Begin to offer a sincere apology for not being the kind of friend you wanted yourself to be and/or the type of friend you could have been. Identify specific things you did that contributed to the downfall and confess them to the other person. Ask for forgiveness, but try not to demand it. Even if your apology is not accepted you can always come back to it later, and try again once you think things through a little more.
Rebuild trust and respect. If the friendship is to be restored or to survive it depends a great deal on how you value yourself as well as the other person (i.e. it depends a lot on the idea of respect.) It is more ethical and more responsible to focus on a person's positive qualities, as focusing on the negative only demeans the value of the others natural humanity, and causes the conflict to be prolonged and reconciliation to be eschewed.