Many times, they can mean the same thing. However, 'peril' is stronger - it means an immediate danger of injury, death or destruction. 'Jeopardy' refers more to the risk of just injury or harm. It can also refer to something less serious than great harm. You will usually hear it as "put something in jeopardy." "The tsunami put the residents of the seacoast town in great peril." "The politician's admissions of wrongdoing have put his re-election in jeopardy."

Basically, they have the same meaning. "Peril" is more dramatic and usually refers to life being in danger. Sometimes it is used figuratively but this is not common. "Jeopardy" is more often used for common objects. There is a popular show called "Jeopardy" where your money might be in jeopardy. If your life is in peril, you are much more likely to die than if your life is in jeopardy. Jeopardy is like walking along the edge of a cliff, peril suggests that you're beginning to fall. "Jeopardize" is a verb that means "to create risk", for example "You are jeopardizing your grades by skipping class." Peril does not have a verb form. I hope my explanation is clear :)