There have indeed been those who had historically referred to themselves as "Illuminati" over the centuries, from the Illuminati of Avignon / Swedenborg Rite, to the Alumbrados / Illuminati of Spain, the Rite of Swedenborg / Illuminati of Stockholm, the infamous Bavarian Order of Illuminati founded by Adam Weishaupt, to a handful of contemporary (and more or less irrelevant) initiatory orders who have the audacity to flaunt the name publicly to attract initiates.

Yet, there exists today a handful of influential groups today who possess "Illuminati" grades within the uppermost degrees of their order, for example: the O.T.O. / Ordo Templi Orientis, the A.M.O.R.C. / Ancient and Mystical Order Rosæ Crucis/ Rosicucrian Order, both of which coincidentally happen to practice goetia and sex magick.

Then you have certain other present-day orders with distinct ties to Illuminism: the Third Order within the Hermetic Order of the Golden Dawn, the S∴S∴ / Third Order of the Silver Star within the A∴A∴ / Argenium Astrum, the Arcana Arcanorum / 90th degree and higher within the A.P.R.M.M. / Ancient and Primitive Rite of Memphis and Misraim / Egyptian (Masonic) Rite, all three of which also curiously practice goeita and sex magick in certain rites of certain grades.

Above those orders, you have key groups whose very name and existence is more often than not denied or obfuscated: the Committee of 300 / Olympians, the Palladium, the Old Black Nobility, Great White Brotherhood, Moriah Conquering Wind, and their Council of 33 and 13 respectively.

Thus, the answer will vary depending on who you ask, and with which faction you are referring. There are some who are hermetic qabbalists, others who are gnostic or christian mystics, druids, thelemites and neoplatonists, many who are hermetic or neo-gnostic luciferian, with some being theistic satanists and many being a combination of the above.

As far as what kind of magic, you have ceremonial magick, solomonic magick / goetic operations, enochian magick, sex magick, tantra, alchemy, theurgy, etc.

My view is this: I respect the right of everyone to practice any religion (or lack they of) they so desire, so long as they do not harm others (i.e. engage in human blood sacrifice) in the process.