What file types should my web designer provide me with?

Your designer should provide you with final png/jpg/gif images if you have agreed that he or she will provide the actual site images. This is a less common case than simply creating the design as a Photoshop comp for the developer to work from.

Final images or not, your designer must supply the PSD(s) so you can have it as a backup and in case you need to make minor modifications. (Why it’s important: Your designer may not always be around. He may be hijacked, hit by a meteor or win “[Country]’s Got Talent” and become a pop star. If anything like that happens, you will be very glad you have your own copy.

I insist that clients receive and archive original artwork files, even if they never plan to use them, because I’ve had too many experiences of picking up a new client — whose original designer is long gone — and finding they have no original logo artwork, no full-resolution images, no usable assets from which to create the new material they are looking for.