The following are major spoilers so make sure you've watched Steins;Gate first if you don't want to kill yourself.
Let's say the original world line is the world line where your grandfather is alive, then you time travel to kill him (to before you and your father were born, of course), instead of causing a paradox (i.e. destroy the laws of physics or everything in the universe, etc.), the world line will just branch from that point of his death in time. How about you? Well... you time-traveled to the past from a different future but you're now in the past so you'd still be there, your original version will no longer be born since it was in a different world line.
There are no multiple world lines. There's only one world line. There are only hypothetical/theoretical world lines that wait for them to be activated. Let's use the Steins Gate world line as an example, all this time it was just a hypothetical world line.. until Okabe with Suzuha time-traveled to the past of the Beta world line and prevented the time machine from reaching Russia and letting it burn in the airplane fire, thus, World War III would no longer occur in the future, and that is in the Steins Gate world line.
