You have three options for removing a pet from adoptable pet searches:
You can click on any of those three buttons next to the pet's name on your "My Pet List" which you can get to under "Pet Lists" in your green navigation bar.
Here's the direct link to your My Pet List:
Hidden pets can be seen only in your admin area on your Hidden Pets list. (Not visible to the public)
Adopted pets can be seen on your Adopted Pets list. (You can check the box on your Adopted Pets list page to make them visible to the public if they have the direct link to your Adopted Pets page, or leave it unchecked so only you can see them.)
Pets that are deleted are permanently removed.