I used the bitcoin calculator to find out on average that it will take me 50 days to generate my first block. What I'm wondering is, if I need to shut off my computer for whatever reason, do I have to start the fifty days all over again once I turn my computer back and start mining again?
Nope, it will just take longer... It is like looking in a pond for a lost coin... the longer that you spend drinking tea, the less time you are looking for the coin. Of course, if you wait too long, the mud will go over the coin and make it harder to find. Meaning that the difficulty will go up, so it takes even longer to find the lost coin.
The key difference between the analogy above and what is really happening with Bitcoin is that the pond is the sea, and the coins are the sand. There are lots of Gold grains evenly spread out in the sand, and you need only to find one Gold grain to win the 'block,' as the difficulty is increased, the size of the Gold grain that you need to find gets larger, so it take more tries.