Foam core is a great material on which to mount photos. It is light and very easy to cut to size, and is handsome enough to serve as a background in its own right, sometimes eliminating the need for a mat.
If you want to frame a photo that has been mounted on foam board, you have a few options.
Using a conventional picture frame and glass, you can simply trim the foam board to size and insert it into the frame as usual, but eliminating the usual layers of cardboard backing provided to snug the picture firmly against the glass. The foam board will do the job. Since it is easy to cut, you can be very creative with the size and shape of frame you choose.
You can also add a mat to the front of the picture to frame it. If you do this in conjunction with a picture frame, you will need to be certain that the frame is deep enough to accommodate both the foam board and the mat board.
Foam board also makes an excellent backing if you want the picture to be in a shadow box frame. The rigid board makes it easy to mount the picture at the rear of the shadow box so that three-dimensional objects can be placed in front of it.
For an informal look, you can leave the mounted picture completely unframed and add embellishments such as braided trim or scrapbook borders directly to it.
The only limitation is your imagination.