Thingiverse First you create an account, then you can download files and customize designs as well as save designs and follow designers.
Pinshape offers designs to download and print. Some sell their designs. Site also includes a blog and tutorials. Creating an account is optional.
Sketchfab allows you to upload and download 3D designs. To download, you must create an account.
YouMagine is an online community where you can download over 10,000 3D designs. An account isn't required to download, but you have to have one to share your images and designs.
Some museums provide digital scans of their artifacts to download and 3D print:
Explore the Smithsonian's 3D digital models, including Lincoln's life masks and the Apollo 11 command module Columbia.
The National Institutes of Health's 3D Print Exchange has models of molecules, bacteria, medical, and anatomical structures to download and print. See also the prosthetics, neuroscience, and heart library collections.
Want to have your design printed, but don't have a 3D printer? With Shapeways you can upload your design, select your desired material, and get a price to have it printed.