What a dilemma! I thank you for writing us here with the question. This might fly in the face of what you have read, but the truth is trying to keep this secret is not going to be easy and is likely to erupt down the road. Since your mom was careless and left her chat open I would tell her you saw it and that her carelessness has put you in a difficult position. Let this be your mom’s issue to deal with. She has made choices that she must now cope with. You not telling her means that you will have to carry the family secret. This is likely to create great stress for you, and most likely will come out in the future.
If you do not feel this is doable I would make an appointment with a family counselor, explain the situation to him or her and get some guidance. You may also make e appointment for you and your mom to go together and talk to her about what you found.
I am definitely not in favor of keeping the secret. It puts you in the middle and creates difficulty for you. It is likely that you will resent your mom and feel uncomfortable around your dad if you try to keep it. Every family event is likely to haunt you, and when the issue comes out (which is very likely –if you found out, your father is likely to as well) it may strain your relationship with your dad if he knows you knew.
If your mom is having an affair there are problems between your mom and dad. By talking to your mom you are putting the issue where it belongs, not taking it on. The find help tab at the top of the page can help you find someone in your area.