A matboard is basicaly to create a matte frame around the image. You'll notice these in frames that are much larger than the actual image (8x10 frame has a 4x6 opening and the space between image and frame edge is the matte).
Stretcher frames are usually used for canvas prints that are stretched over the frame. The image wraps around the frame and is fastened at the back, creating an image with no border at all.
The easiest way to proceed for you is to print the photo at your desired size and then take it with you when you are shopping for the frame. That way you can decide if you want a 12x18 frame that the entire image utilizes, or if you want a frame with bigger dimensions that centers your image within a matte.
Anybody feel free to correct me if I've got it wrong.
