A Google doodle is the picture that is used to show the word Google when you go to the Google website.
Sometimes Google just has the word Google there in colored letters. However on days they consider significant for something they will often have the word Google appear in various shapes expressing that day. The shapes will be significant to what ever they are celebrating.
Sometimes Google has competitions to draw the Google doodle for these occasions.
It is interesting to see what they do with the word Google on various holidays and other significant days
Recently Google starting personalizing the Google doodle. If you have given Google your birthday thru one of their sites (Google, Picasa, BlogSpot, YouTube, etc) then if you go to Google on your birthday and are logged in Google, the Google doodle will be a birthday theme just for you.