Steps

1. "WHY!" - The first thing to understand is that cheating often signifies that there is something wrong, or missing from your current relationship. The goal before you apologize is to find out what that problem is, so once you and your partner move past the initial shock of your actions, you can decide how to best address the problem .
2. "Do you love them?" If you got caught cheating on your significant other, they will be angry, and will most likely ask a lot of detailed questions. These are going to be hard to answer...but be honest. Don't go into dramatic detail over your actions, but be honest about what you feel appropriate to answer. If you shut down now and refuse to answer the questions then a wedge could grow as a result and could lead to further distrust from the other person as well as an inability to communicate and truly fix the problem.
3. Communicate- There isn't going to be any way you can put a bandaid on this situation. In order to fix it, your going to have to reprove your worth, which means dealing with a lack of trust for you over a LONG period of time. Be willing to call  if you're going to be late from a night out  and if your partner asks you be in touch, DO be in touch. If you say that your going to be home by 11...then be home by 11.
4. Understand In situations like this emotions will run deep and strong. By your actions you've torn away a piece of your partner's heart and that scar, like any other, can last for the rest of his life. That doesn't mean you can't salvage the relationship.  Once emotions have died down...talk it out. Explain  that there is no excuse for your actions and that cheating was a bad solution to a serious problem. Explain what aspects of the relationship you're not happy with and see about counseling. Obviously anything you try might not work, and if the decision is made that it can't be fixed...then walk away. Staying in a relationship for the wrong reasons is just as hurtful to both parties as cheating.
5. Cut off communication- Obviously your partner isn't the only one involved in this now that you've cheated. You've taken an outsider into it as well. The person you cheated with. You need to contact this person, under the supervision of your partner, and explain that your actions were wrong. If you wish to stick it out with your partner that is you must cut off all ties to the 'lover'. The hard part to this is often times, the 'lover' is someone close to your life, a co-worker, a friend, the bartender at your local hang out...but you will have to find a way to ignore whatever feelings you have. Again this is only if you're going to continue to try and make it work with your partner.
6. Patience Try counseling and if things don't improve or you both find you hold feelings of resentment towards the other then it's best to cut your losses and split. Not all problems can be fixed in a relationship, and sometimes what you want, and whats healthiest for the both of you are two very different things. Don't set a time limit for how long your spouse is "allowed" to remain hurt or angry. By refusing to answer questions or acting like it's time to move on before your spouse is ready, is not a good plan. You started this process in motion so you must be prepared to deal with the mess your selfishness left behind. Remember, it takes a moment to breakdown trust and a lifetime to re-build it.

Tips

• Don't place blame, or try to justify your actions. Nothing but admitting fault and apologizing, and communicating about the problem will help the situation.
• Make sure you deliver a complete and proper apology to your partner. The apology itself won't remedy the situation but a perfect apology CAN be the first step in the long road to forgiveness.