A manufacturer is a company that's responsible for making or putting together a particular product. For example, Dell is a computer manufacturer. Although Dell doesn't make all of the computer parts, many parts are designed by Dell, and Dell assembles all computers. Product replacements, repairs, and support are handled by the manufacturer of a product.
A company like Dell is an OEM, and responsible for supporting the hardware and software that came with the computer. In other words, if you had a problem with the computer or needed repair you'd call Dell.