I did one semester of Persian language at Iran cultural centre,New Delhi--India.I asked my teacher what was the root of the word Iran.He said Iran comes from Eer {big sound of E followed by a full R}.It means brave.

I guess Iran is plural of Eer.

Iran is actually a very ancient word.  That bit of folklore is pretty interesting in that it parallels a Middle Persian story about a certain Avestan hero named Manuschir who bore three sons; Salm, Iraj, and Turan.   Salm was given the rule of the West and thus was the progenitor of the Romans, Iraj was given the rule of the Center and thus was progenitor of the Iranians, and Turan was given the rule of the East and thus was progenitor of the Turks.  In actual fact, these names came from the Avestan form of East Iranian, the original names being Sairimas, Aryas, and Tiuryas, respectively.  The name Iraj came from "Iran wej" which was originally Airyanem Vaejah of the Avesta, the supposed "home of the Aryans".  Hence we already knew that in Avestan times (i.e. before the Median and Achaemenid times) the ancient Iranians knew of an ancient land Airyanem Vaejah, probably located to the northwest of present-day Iran which subsequently redefined to mean the whole of Iran itself.  The ancient Greeks knew Iran as "Ariana" and the Sassanids knew Iran as "Eran" or "Eranshahr".  Thus it is not difficult to see that the name "Iran" is the modern form of the very ancient term "Airyanem", (i.e. the "land" of the Aryans).

Herodotus recorded a tradition from the Medians themselves which states that they originally called themselves Arioi, (i.e. Aryans), and the Persian king Darius in his inscriptions called himself "an Ariya of Ariyan seed" and was said to have written "in Ariyan", thus declaring that he and his people were "Iranians".  In other words, the ancient Persians and Medes saw themselves as Aryans.  "Iranian" is simply the modern form of "Aryan".