Origins
The prefix 'inter-' is a highly productive Latin-derived prefix in English, used to form words indicating 'between,' 'among,' 'mutually,' or 'reciprocally.' It descends from Latin 'inter' (between, among), which in turn comes from Proto-Indo-European *h₁n̥ter or *enter, a comparative form of the locative root *h₁én (in). The same root produced Sanskrit 'antár' (within, between), Greek 'éntera' (ἔντερα, intestines, literally 'inner parts'), Old Irish 'eter' (between), German 'unter' (among, between), and English 'under' — where the sense has drifted from 'between' to 'below.'
In Classical Latin, 'inter-' was productive in verbs and nouns. Verbs: 'intercedere' (to go between — whence 'intercede'), 'interrumpere' (to break between — whence 'interrupt'), 'interferre' (to carry between — base of 'interfere'), 'intermittere' (to send between — whence 'intermit'), 'intervenire' (to come between — whence 'intervene'). Nouns: 'intervallum' (space between palisades — whence 'interval'), 'internus' (inward, inside — whence 'internal'), 'intestinus' (inner — whence 'intestine').
'Inter-' entered English in the fourteenth and fifteenth centuries in many learned borrowings from Latin and Anglo-Norman French: 'intercede,' 'intercept,' 'interchange,' 'intercourse,' 'interdict,' 'interest,' 'interfere,' 'interior,' 'interject,' 'interlude,' 'intermediate,' 'intermission,' 'internal,' 'international' (much later — 1780, coined by Bentham), 'interpose,' 'interpret,' 'interrogate,' 'interrupt,' 'intersect,' 'interval,' 'intervene,' 'interview.'
Middle English
Some 'inter-' words came via Old French 'entre-,' which is the regular French reflex. These preserve the French form in English: 'enterprise' (Middle English from Old French 'entreprise,' a taking between), 'entertain' (Old French 'entretenir,' to hold between / keep up), 'entreaty' (from 'entreat,' Old French 'entraitier,' to treat between). These three words contain the same prefix as 'interrupt' and 'international' but are not easily recognised as such because they came through the French-reduced form.
From the seventeenth century onward, 'inter-' became one of the most productive English prefixes, freely coining new words on both Latinate and native bases. International diplomacy and politics produced 'international' (1780, Jeremy Bentham), 'intercontinental' (1850s), 'interstate' (1845, American), 'interurban' (1883), 'intergovernmental.' Science and technology produced 'interaction' (1832), 'interface' (1882, originally a surface between phases; popular in computing from the 1960s), 'intercom' (1940, abbreviation of 'intercommunication'), 'internet' (1974, from 'inter-network'), 'interactive' (1832 in general sense, 1960s in computing), 'interlude,' 'interlock,' 'intermix,' 'interplay.' Sport, social life, and entertainment produced 'interleague,' 'intercollegiate,' 'intermarriage,' 'intermingle.'
A closely related Latin prefix is 'intra-' (within, inside), which contrasts minimally with 'inter-' (between): 'intramural' (within walls / within an institution) versus 'intermural' (between walls / between institutions); 'intranet' (network within an organisation) versus 'internet' (network between organisations). The Latin pair 'inter / intra' remains semantically active in English scientific and technical vocabulary.
Proto-Indo-European Roots
A second minimal pair is 'inter-' (between) versus 'intro-' (into, inward), as in 'introduce' (lead into) versus 'interject' (throw between). Both come from the same PIE root family but developed specialised senses in Latin.
Phonologically, 'inter-' is generally pronounced /ˈɪntə/ in British English and /ˈɪntər/ in American English, with the 't' often reduced or voiced intervocalically in American English (the familiar 'innernational' pronunciation of 'international'). The prefix is normally stressed as the primary stress of the word when transparently added, or is reduced when the word is fully naturalised.
In scientific and technical vocabulary, 'inter-' systematically marks reciprocity or in-between-ness. Biology: 'interspecies,' 'interbreed,' 'interdependence.' Mathematics: 'interpolation,' 'intersection,' 'interval.' Computing: 'interface,' 'interlink,' 'interrupt,' 'internet,' 'interoperable,' 'interprocess.' Chemistry: 'intermolecular,' 'interionic.' Medicine: 'intercostal' (between ribs), 'interstitial' (in the space between).
Latin Roots
Allomorphy is modest. 'Inter-' generally does not assimilate to the following consonant, unlike Latin 'in-' or 'sub-.' Minor assimilated forms exist in Latin: 'intellegere' (to perceive, from 'inter-' + 'legere,' literally to choose between — whence 'intelligence,' 'intellect') is occasionally analysed as showing 'inter-' reduced to 'intel-.' This assimilation is historical, not productive in English.
Representative 'inter-' words include: interact, interaction, interactive, intercede, intercept, interchange, intercom, interconnect, intercontinental, interdependence, interdisciplinary, interest, interface, interfere, interim, interior, interject, interlinear, interlock, interlocutor, interlude, intermediate, intermediary, intermezzo, intermingle, intermission, intermittent, internal, international, internet, interpersonal, interplanetary, interplay, interpolate, interpret, interpretation, interracial, interrelate, interrogate, interrupt, intersect, intersection, intersperse, interstate, interstellar, interstitial, intertwine, interval, intervene, intervention, interview, interweave. Via French entre-: enterprise, entertain, entreaty, entrée. Related: intercostal, interoperable, intercessor, interceptor.