The Persian Empire included all of what is now Iran, and in fact Persia was the official name of Iran until 1935. At its height about 500 BC, the founding dynasty of the empire, the Achaemenids, had conquered Asia as far as the Indus River, Greece, and North Africa including what is now Egypt and Libya.
The beginning of the Persian empire is set at different times by different scholars, but the real force behind the expansion was Cyrus II, aka Cyrus the Great, in the mid-sixth century BC. Cyrus was part of the Achaemenid Empire, with Cyrus’s capital first at Hamadan and then Pasargadae. In 330 BC, Macedonian Greeks led by Alexander the Great overthrew the Achaemenids and established what was called the Seleucid Empire after Alexander’s general. The Seleucids were in turn followed by the Parthian and Sassanid dynasties. The Sassinids were defeated by Arab caliphs in the mid-7th century AD, and by 651, the Persian empire was ended.