The Google Doodles are created and drawn by a team of illustrators and engineers that work for Google. The illustrators are known as Doodlers. Their names aren't usually made public, but in fact the original idea for Doodles goes back to the company founders, Larry Page and Sergey Brin. Having created the main logo, before presenting it at a business event they started playing with it, creating a simple stick drawing to go with the logo. It looked good and so the idea of having lots of different doodles around the logo was born. Since then Google's engineers and artists have created over 1000 doodles around the world.
Members of the public are also invited to send in their own ideas for a Doodle. The company don't guarantee a reply as they get thousands of emails every day, but they say they are very happy to receive the ideas. (Well, they would be - a good idea would benefit them of course).
You can find the story of the Doodles, as well as email details if you want to send one in, here.