An infix is a simple affix that is inserted into a morpheme (cf. Helmbrecht & Lehmann 2008, section 3). Here is a stock example from Latin conjugation (the present is formed by infixation):

The above definition requires the following comments:

• Taken literally, an affix (ad-fix) should be attached at the periphery of its host, so that an infix could by definition not be an affix. This is, however, not meant by the term ‘affix’; it means just any bound non-root.
• A simple affix is distinguished from a discontinuous affix. There are also discontinuous counterparts to infixes; but they are called transfixes.
• An infix is by definition inserted into a morpheme. The host morpheme is generally a root. However, an infix in an affix is not excluded by the definition; an example is offered below.
• An infix disrupts its host, making it discontinuous. An affix that follows the root, e.g. the -ly in lone-ly-ness, is by definition not an infix, but a suffix. (There are non-peripheral suffixes!) This is so despite the fact that non-peripheral suffixes are again and again called infixes (e.g. Buccellati 1997:77); and the same goes for prefixes (e.g. Wikipedia s.v. Altirische Sprache).

Diachronically, an infix can originate in two ways, either by metathesis or by entrapment.

• In metathesis, an outer affix crosses the syllable boundary of its host and becomes its infix.
• In entrapment, an affix that is flanked by two other morphemes becomes their infix in the moment that these two are united into one morpheme.

In 99% percent of the cases, the host morpheme of the infix is a root. However, by entrapment, it is even possible to get an infix into an affix. Here is an example from Gedeo (Highland East Cushitic), from Gragg 2008:

What appear synchronically to be monomorphemic 1^st and 2^nd singular suffixes were originally composed of two morphemes, the first marking person and the second marking aspect. The negative suffix attached to the first of these, and then followed the aspect suffix. Later the person and aspect complex was monomorphemicized so that the negative suffix got entrapped and became an infix.

References

Buccellati, Giorgio 1997, "Akkadian." Hetzron, Robert (ed.), The Semitic languages. London: Routledge (Routledge Language Family Descriptions); 69-99.

Gragg, Gene 2008, "Morphological expression of verbal negation in Cushitic". 3^rd Meeting of the International Association of Comparative Semitics, Turin, 2 - 5 october 2008.

Helmbrecht, Johannes & Lehmann, Christian 2008, "Hocank's challenge to morphological theory." Harrison, David & Rood, David & Dwyer, Arienne (eds.), Lessons from documented endangered languages. Amsterdam & Philadelphia: J. Benjamins (Typological Studies in Language, 78); 271-315. [ download ]