Steve Jobs had worked during the summer at an apple farm, and admired the Beatles' record label, Apple. He also believed apples to be the most perfect fruit. He and Steve Wozniak were trying to figure out a name for their new company, and they decided that if they couldn't think of one by the end of the day that was better than Apple, they would choose Apple. They couldn't think of anything better, so on April 1, 1976, Apple Computer, Inc. was born.

 
But they needed a logo. The first design included Sir Isaac Newton, a tree and a banner that said "Apple Computer." Jobs decided they needed a less busy logo, one that would signify a brand. The second logo attempt was very similar to the current logo, but without the bite taken out of it. Jobs thought this logo looked too much like an orange. The third attempt was the logo that Apple still uses.
 
Taken from Apple-History's FAQs