The Persian army

Conception
It is likely that the Persian army was originally made up mainly of nobles and their immediate bandaka, in early feudal fashion.  Herodotus' famous summary of Persian education (1.136) makes sense only in the context of a military aristocracy, as does Xenophon's more detailed depiction in the Cyropaedia (1.2).  Per Herodotus, Xenophon and Strabo, the sons of nobles were required to serve in the military.  Kings and nobles took part in (and sometimes died in) fighting at the front line.

Land value was sometimes assessed according to feudal underpinnings:  In the Achaemenid Babylonian texts, royal allotments are called bīt qašti, "bow land," bīt sisī, "horse land," and bīt narkabti, "chariot land," probably referring to the valuations of land needed for the support of an archer, a horse and cavalryman, and a chariot and crew.  (Per Erich Ebeling and Bruno Meissner, Reallexikon der Assyriologie und Vorderasiatischen Archäologie, most holders of these estates during the Achaemenid period were not in fact soldiers, and if they were required to provide military service, probably hired soldiers.)

The foreign service
Much of the very large armies employed at various times, such as Xerxes' invasion of Greece, consisted of levies from various subject nations.  They were dressed and armed in their native manner, but led by Persian commanders.  Briant, however, has argued that most of the national contingents at this particular gathering weren't intended as a fighting force but as an imperial review and display of power.  The contingents that Herodotus actually mentions in battle are only a handful of those who turned up.

In the late 5th and 4th centuries, large numbers of Greek mercenaries entered Persian service.  The hoplite panoply and phalanx tactic were difficult to match in head-on combat with anything other than themselves.  Xenophon decries this situation (Cyr. 8.8) as part of, in his opinion, an overall decline in the martial character of the empire.

Troop types
Such a criticism would be most sensible, of course, if the Persians had ever had a tactic for meeting the phalanx head-on.  There are hints that they may have tried to develop such a thing later:  The mysterious kardakes at the Battle of Issus are described by Arrian, ostensibly quoting Ptolemy who was a witness to he battle, as "hoplites," and modern writers have sometimes taken this to mean that they were a Persian adaptation of the Greek infantry type.  From the Alexander Sarcophagus it appears that Persians did occasionally use Argive shields.  Strabo, however, says that the kardakes were young men training for war in the king's service.  In any case, I know of no evidence of Persian hoplites in the early 5th century.

The heaviest infantry contingent at that time was the Immortals.  (I won't repeat the well-known tale of the name's origin except to say that it is disputed - Lendering has ventured that it is a confusion with a similar Persian word meaning "companions," or in other words, the king's personal army/bodyguard.)   These likely carried the full panoply described in the Histories 7.61, including scale corselets and the short akinakes sword.  Despite being, ostensibly, the best Persian soldiers, even they lacked the ability to meet a phalanx head-on, probably due to using an open-order formation and shields unsuitable for the close-order shoving that characterized the phalanx.

For the most part, the Persian army relied on a more static line infantry, light infantry and cavalry.  The line infantry were armed with spears and carried large rectangular shields, which formed a shield wall to protect large numbers of archers.  These, the spearmen and archers, were likely the largest Persian contingents in the Graeco-Persian wars.  From Greek art in the late 5th and 4th centuries Asiatics are shown with smaller crescent shields and spears or javelins.  Slingers were also known.

The cavalry were likewise light cavalry, relying on bows and javelins (which could double as short spears).  By Xenophon's day, at the earliest, they were rather heavier-armored than the typical infantry (likely because they were wealthier, or were furnished by wealthier estate-holders), and the horse also had some armor.  But they lacked stirrups, which would have allowed a proper charge in the manner of heavy cavalry.

Chariots had lost much of their tactical value in the Iron Age, when warriors started riding horses directly, but were still considered a status symbol.  Late in the 5th century, the Persians started employing scythed chariots, but these failed to have much effect at the major battles of Cunaxa and Gaugamela, as well-trained infantry could sidestep them.  Only when catching men unprepared, as in one incident during the early 390s war between Spartan king Agesilaus II and the satrap Pharnabazus of Lydia, did scythed chariots prove useful.

Divisions
Persians appear to have had a liking for decimal division, per Herodotus (7.81):  The names for subdivisions are commonly given as daθabam (10, commanded by a daθapati or daθapatiš), satabam (100, satapati[š]), hazarabam (1,000, hazarapati[š]) and baivarabam (10,000, baivarapati[š]).  I have yet to track down the original sources for these terms.

Next up:  You can't use a bow without it.