If you want to download a model to 3D print it on your own printer or a printing service like Shapeways, here's a checklist.
Some files are ready to print right away.
However, because you can only download the author's original files, some models will need a bit more work before printing.
3D Printing Checklist
These points may vary between different printers, so it's always best to check with the manufacturer!
File format - What kind of file format does your printer require? Most printers will accept STL files, although commercial services can work with a wider range of file types (e.g. full color printing, see below). Most 3D applications can export to STL so it's easy to convert a model that you downloaded from Sketchfab.
Watertight - Is your model 'watertight'? 3D printers get confused if there's a 'leak' in your model - essentially this would reduce the model to a two-dimensional shell. So take care to close your model! Of course you can still have models with holes in them, like a donut, but make sure that the surface is continuous and closed.
Manifold - Related to the concept of watertightness is 'manifold'. This is a mathematical term describing object surfaces. Basically, it confirms that the surface can exist in reality - the main requirement being that each edge should be connected to exactly two faces. Most 3D apps have support for testing manifoldness. If yours doesn't, try the free Meshlab software.
Wall thickness and Model size - Does your model conform to your printer's minimum thickness (the thinnest part it can print), and will it fit in the print area? Each printer has its own minimum and maximum model sizes, be sure to check them!
Overhang - This is a bit more technical. Some printers, especially FDM printers (like Ultimakers or Makerbots), have a limit to the 'angle' of some of the faces. A large overhang will require adding support material. Again, check with your printer to see if you can add supports.
Full color support - Most printers these days print in one material. Only a few can handle full color like the ZCorp or Visijet printers. For 'monochrome' printers, the STL file is fine. For full color, you'll usually have to provide a VRML file with the required vertex colors or UV maps.
Polygon threshold - the maximum number of polygons the printer can handle. Keep the printer resolution in mind when deciding on the polygon count. Details that are below the resolution won't get printed anyway, so don't bother sending them to your printer. A good rule of thumb is to stay below 1 million triangles. You can use mesh decimation software to remove unneeded details while still preserving your model's shape.
Price - 3D printed models can get expensive fast! Most printing services base their price on the printed volume. As such, increasing your model's size by a factor of two will result in an 8x increase in price. The good news is that you often only need a small size adjustment to make your print match your budget.
Resources and Tutorials
Photoshop allows you to prepare files for 3D printing.
Shapeways offers software tutorials for preparing 3D printed character models.
Free software for file conversion: MeshLab
Autodesk MeshMixer is a great tool for creating mashups of existing models. And it's free!