Prefixes and suffixes are both kinds of affixes. That is, they are word parts that attach to the beginning or end of a word or word base (a word stripped down to its simplest form) to produce a related word or an inflectional form of a word. Examples are in- in informal and both re- and -ing in reporting.
A third kind of affix is called an infix. Infixes are inserted into a word or word base. English uses very few infixes, but a couple examples are the plural-making s in words like cupsful and passersby, and various swear words, like damn in informal constructions like guaran-damn-tee.
A combining form is a form of a word that only appears as part of another word. There are a number of kinds of combining forms, each classified by what kind of word results when the form is used. For example, -wise in clockwise is an adverb combining form; -like in birdlike is an adjective combining form; -graph in photograph is a noun combining form; and -lyze in electrolyze is a verb combining form.