If looking at their etymology, I can roughly get the following ideas:
democracy=people rule
autonomy=self management
economy=family management
anarchy =no rule
hierarchy =high priest rule

I think "cracy", "nomy" and "archy" have the similar meanings.

So why do some words end in cracy, others in nomy, and still others in archy?

Thanks very much.
Words can come and evolve from different root words.
There is a difference between managing and ruling.

English has many root words - the majority being from Proto-Indo-European (PIE), Old German (Anglo-Saxon), Old French, Latin, Greek. and Breton.

Here is some etymology on the words you listed: (source: http://www.etymonline.com )
1570s, from M.Fr. démocratie (14c.), from M.L. democratia (13c.), from Gk. demokratia "popular government," from demos "common people," originally "district" (see demotic), + kratos "rule, strength" (see -cracy).

comb. element forming nouns meaning "rule or government by," from Fr. -cratie or M.L. -cratia, from Gk. -kratia "power, might; rule, sway; power over; a power, authority," from kratos "strength," from PIE *kratus "power, strength" (see hard). The connective -o- has come to be viewed as part of it. Productive in English from c.1800.

1620s, of states, from Gk. autonomia "independence," noun of quality from autonomos "independent, living by one's own laws," from auto- "self" (see auto-) + nomos "custom, law" (see numismatics). Of persons, from 1803.

economy (n.)
1530s, "household management," from L. oeconomia, from Gk. oikonomia "household management, thrift," from oikonomos "manager, steward," from oikos "house" (cognate with L. vicus "district," vicinus "near;" O.E. wic "dwelling, village;" see villa) + nomos "managing," from nemein "manage" (see numismatics). The sense of "wealth and resources of a country" (short for political economy) is from 1650s.

1530s, from Fr. anarchie or directly from M.L. anarchia, from Gk. anarkhia "lack of a leader, the state of people without a government" (in Athens, used of the Year of Thirty Tyrants, 404 B.C., when there was no archon), noun of state from anarkhos "rulerless," from an- "without" (see an- (1)) + arkhos "leader" (see archon).

one of the nine chief magistrates of ancient Athens, 1650s, from Gk. arkhon "ruler," noun use of prp. of arkhein "to rule," from PIE *arkhein- "to begin, rule, command," a "Gk. verb of unknown origin, but showing archaic Indo-European features ... with derivatives arkhe, 'rule, beginning,' and arkhos, 'ruler' " [Watkins].

mid-14c., from O.Fr. ierarchie, from M.L. hierarchia "ranked division of angels" (in the system of Dionysius the Areopagite), from Gk. hierarkhia "rule of a high priest," from hierarkhes "high priest, leader of sacred rites," from ta hiera "the sacred rites" (neuter plural of hieros "sacred;" see ire) + arkhein "to lead, rule" (see archon). Sense of "ranked organization of persons or things" first recorded 1610s, initially of clergy, sense probably influenced by higher.
Thank you for your help.
It is very informative.