Sam's theory (which we learn in a later episode comes from a scientist who he helps by leaping into his assistant)...
Your life is a string - starts at the beggining - the time you are alive is the length and at the end of the string is the end of your life. Now, if you could crumble the string up, each point in your life would touch another point in your life.
Project Quantum Leap was supposed to let the person travel through time (but limited to the user's own life time).
However, the problem with the project is that it left Dr. Sam Becket into OTHER people (though people who were alive in his life time - with a couple in-show exceptions for certain reasons). He then goes from life to life, striving to put right what once went wrong in the original time line.
His only guide is AL, someone fom Sam's "present" . Al works in a holographic chamber that let's him see where Sam is and the two can communicate that way. No one else can see Al.
I think that certain things in that show really rocked.
(not that it didn't have flaws, but I still liked it)
As has already been said, the first episode involved Sam using the Quantum Leap accelorator (the name of the machine that powers Project Quantum Leap) prematurely.
Thus, he can't really control his leaping. All he and Al have been able to figure out is that by fixing things that went wrong Sam "completes" what he is supposed to do and ends up leaping again.
Also, Quantum leaping leaves Sam's brain like swiss cheese - he has trouble remembering things sometimes (esp. about his own life).
Some of the most amusing moments are when he interacts with "side characters" who aren't part of the particular story line...
for instance, he had a random encounter with a doctor who was choking. Sam does the Hiemlich (sp?) manuever on the Doctor and saves his life. Someone later says to the doctor, "Dr. Hiemlich, are you alright?"
similiar story with a young steven king and others ...