it's something that you say in order to tell someone with a lot of force that you will not do something 'Would you kiss him?' 'Not on your life!' or *Not if your life depended on it!*
"Not on your life!" dates from the eighteenth century. It refers to the old belief that you swore "on" someone's life, and if you didn't fulfill your oath, your life was forfeit.
Similar (and some more polite ways to say this are:
In a word, no.
Not on your life.
Over my dead body.
Count me out.
I'd rather not (if you don't mind).
I'd love to, but...
The British seem to have an equivalent in "not on your nelly!"