If you’re running a logo design contest, you should ask for a vectorized image in either of the two file formats listed below (note that you may need to download a special program to view these files):
You may also want to ask the designer to provide a JPEG, GIF, or PNG file so that you can view the design without downloading special software.
For website designs, you should ask for a layered Photoshop PSD file.
Don’t ask the designer to provide any of the files for the font they used in your design, since font licenses are usually non-transferrable. But do ask the designer for the name of the font so you can buy a license if you need to.