The simple answer is that vector art (.EPS, .AI, sometimes. PDF) can be infinitely resized larger, with zero loss of quality and zero pixelation.
JPG, TIFF, or PNG files cannot be resized larger without pixelation (the imagery becomes “blocky”), which results in very bad looking prints.
Vector files are the best, and depending on the process & product being imprinted, are required. Vector files are .EPS or .AI, and sometimes .PDF. (PDFs can contain both vector and non-vector artwork, so ensure it’s vector, or we can check for you.)
.JPG, .PNG, .TIFF files are raster files, but depending on the process and the product, they will work– provided that they are high-resolution (large files).
Photos: Photos that are to be imprinted should be high-resolution .PNG, .JPG, or .TIFF files, at least 300dpi at the size you’re needing printed.