As Frances Yates demonstrated in various books Rosicrucianism is more a cultural phenomenon and esoteric current than a movement. Its main componenents being Hermetic Gnosticism, rational philosophy, liberal Protestant Cabalism, science, alchemy, the healing arts and sometimes Magick. In simpler terms it was the trend of combining modern rational science with a Christian interpretation of the ancient Hermetic traditions. A trend found in early scientists as late as Isaac Newton.
These ideas first moved towards a synthetic body of doctrine in the Duchy of Frederick of Wurttemberg, who sponsored a small cultural circle developing these notions in 16th Century Germany (as did the school around Frederick V of the Palatinate in the following century). The most developed philosophy to emerge from this was the Pansophism of Simon Studion (who was much influenced by the ideas of Joachim of Fiore, the first 'New Ager', and the Elizabethan magus John Dee). However the idea of an organised movement actively promoting these ideas is now believed to have been first initiated by a small esoteric society, calling themselves the Brethren of the Rose Cross, at Tubingen University in Wurttemberg in 1608.
This group included Johann Andreae (Xtian mystic, alchemist and scientist), Christoph Besold (multilingual scientist and Hermetic Cabalist), Tobias Hess (Hermetic Pansophist, Paracelsian and mystic), and the Campanellans, Tobias Adami, Wilhelm Wense, and Abraham Holzel. In addition to Pansophism, the ideas of the Italian Hermeticist Campanella were very influencial on this group (including his political aims of creating a new world order based on a Hermetic theocratic utopia, in his work the 'City of the Sun'). The group was also under the spell of an early manifestation of German unification and Nationalism. The Rose Cross motif is variously attributed to the red cross of the Templars, and/or the cross merged with the mystical rose, of Aphrodite and/or Sufism (none of which is very convincing!).
This group did not attempt to create an international organisation though, instead they seem to have adopted what would be later known as the 'Rosicrucian Ploy', the publication of a cryptic manifesto and the announcement of a secret society, which proved impossible to contact. A now classic trick which forces the Rosicrucian wannabe to form their own group instead and try to recreate the original. Thus no body of dogma is passed down and free thought predominates. The trick worked, amongst those known to have attempted, and failed, to contact the Fraternity was the influencial philosopher Renes Descartes (thus forever installing an unfortunate Cartesian bias in many subsequent Rosicrucian Orders), Rosicrucianism (and conversely paranoia about Rosicrucians, 'the Invisibles') spread like wild fire across Europe.
Of the original group it appears to have broken up. Some, like Besold, its real driving force, converted to Catholic mysticism and withdrew from the world, others became refugees from the Counter Reformation, and its religious wars, in the 'Christian Unions', while some may have disappeared into other more traditional orders.
Three currents of Rosicrucianism emerged:
The first, that associated with Germans in the refugee unions was centred on science, and in England partially reformed as the Invisible College, merging with developing Freemasonry, into which it was absorbed, and eventually founding the Royal Society.
The second consisted of the alchemical Rosicrucian Fraternities (such as the Golden and Rosy Cross tradition) which emerged from established associations of alchemists (such as the Order of Inseperables, a group sponsored by the mining industry, into which some early Rosicrucians may have fled). These were more conservative and preserved a nationalistic tendency. Some would later form alliances with continental Rose Croix Freemasonry.
But the largest current was that of of those autonomous orders that proliferated after the Fama and other manifestos were published.
Rosicrucianism prompts a variety of responces from other Occult groups, Traditionalists see them as 'modernisers' who corrupt ancient traditions and have no respect for real Magick, while the Illuminati camp see them as unreformed conservative mystics still trapped in religion and superstition.
Today a vast variety of Rosicrucian Orders exist, some public (like AMORC) some secret (such as ??).