Assuming you still want to know when items are listed matching your search - all you need do is edit your alert to stop receiving emails.
Sign in, go to 'My Account' click on 'My Alerts' and then on the 'Edit' button next to the relevant alert. Go through the create and save alert process and at the save stage change the email frequency in Step 2 to 'Never'. Any items StuffAlert finds matching your search criteria will still be automatically addded to your 'My Found Stuff'.
If you want to keep your search but stop receiving email alerts when new items are listed simply pause your alert. Sign in, go to 'My Account' click on 'My Alerts' and then on the 'Pause' button next to the relevant alert. To restart a paused alert in 'My Alerts' click on the 'Activate' button next to the relevant alert.
You can of course delete an alert quickly and easily by ticking the box next to the relevant alert and clicking on the 'Delete' button at the bottom of the page.