Islamic means relating to Islam; it is a synonym of the adjective Muslim and is best used in the context of religion. Use Islamist rather than Islamic when referring to a political ideology that purports to be based on Islam. Do not refer to Iran as the Islamic Republic. Afghanistan, Mauritania and Pakistan are also Islamic republics; many other countries make reference to Islam in their constitutions.
Islamism encompasses a wide range of views—from the constitutionalist Justice and Development Party in power in Morocco, to Egypt’s ousted Muslim Brotherhood, the Hamas movement in Gaza, al-Qaeda and the gore-loving Islamic State. At the moderate end of the spectrum you can talk of political or moderate Islamists. (Members of Tunisia’s Ennahda now want to call themselves Muslim democrats.) At the more extreme end you can talk of radical, militant or even violent Islamists. For those who exalt violent jihad as a core belief, eg, al-Qaeda, call them jihadists (see details below).
Avoid referring to Islamic State with the definite article. Its ideology is an outlier in the world of Islam. You may, if you want, qualify its title in some way, eg, the jihadists of Islamic State or the Islamic State group. Similarly, qualify the short-lived entity it created in Iraq and Syria in 2014 as a “caliphate” (inverted commas needed) or would-be caliphate.
Jihad is the Arabic word for striving. For modern Muslims, it may mean military war to propagate Islamism, ie, to spread Islam as a religious, political and social ideology (jihad of the sword). Or it may mean spiritual struggle for personal purification and moral betterment (jihad against oneself). Or it may merely mean doing right, improving society and being virtuous (jihad of the tongue or of the hand). A religious obligation for all Muslims, jihad is for most a non-violent duty. Do not therefore use it simply to mean holy war. Rather make clear what sort of jihad is under discussion in the context.
Someone engaged in jihad is a mujahid (plural, mujahideen) or a jihadist (prefer to jihadi). In practice, these terms nowadays are always used of Muslims engaged in an armed struggle, though mujahideen may simply be Muslim militants fighting for a cause whereas jihadists are always fighting to spread Islamism by force.