At the bottom of my screen there is the recycle bin. Is there someway to unpin it from there?
By the way, where is the recycle bin in Finder?
It's not possible to unpin the Trash, and it isn't readily available in Finder.
The Trash is aggregated from the files stored in a hidden .Trashes directory in the root directory of each mounted volume.
Because of that, you can't add it to Finder as a regular folder, because it just isn't. You could probably get a similar behavior by creating an AppleScript application:
Open AppleScript Editor, enter the following and save as application:
Add to the Finder's sidebar. It'll show the Trash when simply clicked, and move all items dropped onto it to the Trash.
It is possible to remove the trash can from the dock. Although it sounds like it's not a good idea if the machine has non-privileged users (ie, users who are NOT Administrators).
EDIT: Be careful. Following this hint may not be the best idea.