Double-clicking would simply open the Spotlight menu, then immediately close it. Try single-clicking the Spotlight icon, and a blue text menu should appear where you can type your search query. You can also press Command-Spacebar from anywhere within Mac OS X and that same blue menu will appear, allowing quick searches using only the keyboard.
You can also hit Command-F in the Finder, and a "Spotlight Find" window will open where you can specify criteria to search by.