This information is probably all over the place already, but for easy reference, I wanted to type it up for those who had not seen it.
Excerpt: This article explains a method for displaying a WordPress search form without a search button. The user simply enters they text they would like to search for, and hits enter. While this is easy to do with the use of a hidden submit button, this method also allows for the use of a default message in the search box (such as “Enter your search”), which keeps you from having to display a label next to the form explaining what it is for. This default message automatically clears when the user clicks to start typing, and puts itself back if the user clicks away. This allows you to have a search form that looks nice and clean, and is still just as functional.