Strangely, you cannot delete your Picasa web account without deleting your entire Google account, which includes Gmail, Youtube and others. To get rid of your Picasa account only, Google suggests the following (Taken from their FAQ, http://support.google.com/picasa/bin/answer.py?hl=en&answer=49106):
Sign in to Picasa Web Albums at http://picasaweb.google.com using your Google Account username and password.
Select the album you’d like to delete.
From the Actions menu above your photos, select Delete album.
Click OK to confirm that you want to delete the album.
Repeat these steps for each album in your account that you wish to delete.