Should you give them a second chance after cheating?
While the answer to this question depends entirely on the situation, the mere fact that you're here indicates that you're considering it, so firstly let's talk about a few instances in which a second chance after an affair is ill-advised.
They compulsively lied to you
If they compulsively lied to you and snuck around behind your back, while keeping you in the dark about their activities, even when you confronted them point blank, you should definitely not give them a second chance. Why? Because they have no respect for you. A person who respected you would not try to make you doubt your instincts or lie to your face while making you feel stupid and/or crazy for addressing the issue. There is something innately cruel to lying to a person's face and then accusing them of being invasive and paranoid for figuring out the truth. If that was the case with you, and you're even considering taking them back at this point you need to seriously consider the notion that you're abnormally attracted to parasitic people who don't have any respect for you.
They're going to do it again
If you feel like they're going to do it again, and you don't want to suffer a repeat of the same infidelity twice, then you're wasting your time with a second chance. Obviously, a second chance after cheating, requires a belief that it was a one-time affair and won't be repeated. Generally, it will, and if you can't deal with that fact, then you're better off ending the relationship right there.
If you want to give them a second chance, here's what to do
Address the root cause of the affair
There are several good reasons to give a spouse a second chance after an affair, but only by addressing the root causes and empathizing with the cheater can you truly begin the healing process. One common reason why otherwise loving and engaged spouses cheat in their marriage is sex addiction. Despite what you may believe, sex addiction is a legitimate psychological issue that has many of the same features as other addictions. Firstly, those who suffer from sex addiction feel ashamed and guilty directly after having sex outside of their marriage, and yet feel compelled to do it. There are support groups based on the 12 step model for those suffering from sex addiction and help is available from counselors and psychologists to break free of the cycle that perpetuates it.
Find a marriage counselor
That brings us to our next point: finding a marriage counselor. If you're serious about giving them a second chance and not just sweeping the whole thing under the rug, then you're going to want to find a good marriage counselor to work through the issues in your relationship that caused the affair in the first place. If both partners are committed to working out their difficulties and saving their marriage, this should be a no brainer. A lot of times, marriage counselors are good sounding boards for learning how to effectively communicate with one another. Particularly, they are good for individuals who still love one another, and yet feel hurt and resentful of behavior that happened in the past. If the marriage is going to be save, then it will require work, and marriage counselors will help with that.