OK, please correct me if I have the heirarchy wrong. Every LDS male can become an Aaronic priest, right? Some are Melchizedek priests, correct?
There are two facets to priesthood in the Mormon church. The first facet is "priesthood office", which more or less reflects one's stage in life; and the second is "calling", which reflects what the person actually does
in the Church.
Basically, there are two orders of priesthood--Aaronic and Melchizedek--and within each of those "orders" are several "offices":
A group of men holding the same office is called a "quorum", and one hour of each Sunday service is devoted to quorum meetings (women go to "Relief Society" during this time). Each ward (local congregation) has a deacons' quorum, a teachers' quorum, a priests' quorum, and an elders' quorum. Each stake (group of 8-10 wards) has one "quorum" of High Priests; on Sundays, high priests in a single ward will get together in a "group". (I'm not aware of any formally organized quorum of bishops or patriarchs in the Church. There are also quorums of Seventies and Apostles, but these are governing/administrative quorums over the entire Church.)
The typical rite of passage for a Mormon male is that as a teenager he spends a couple years at each of the first three offices of the Aaronic Priesthood (Deacon, Teacher, Priest), and at 18 or 19 years old he is ordained to the office of an Elder in the Melchizedek Priesthood.
There are a multitude of "callings", or roles, that one can fill in a local congregation (teacher, librarian, nursery leader, etc); and lots of them don't require that the person have any office in the priesthood at all. But the main positions that involve running actual congregations, or groups of congregations, are as follows:
A ward is supervised by a leader, called the "bishop", who has two counselors (collectively called the "bishopric"). The bishop must be ordained to the office of a bishop in the Aaronic Priesthood, and to the office of a high priest in the Melchizedek Priesthood. He will retain these "offices" even after his term of service ends, and on Sundays he will meet with the high priests group. The bishop's counselors are also ordained to the office of High Priests, but not to the office of a bishop. Again, they retain that office even after their term of service is over. But as a matter of secular law, they are no longer clergy.
A small congregation is called a "branch" and is overseen by a branch president and his two counselors. A branch president must be ordained to the office of elder in the Melchizedek Priesthood. Elders who are serving in branch presidencies, AFAIK, are recognized as clergy under secular law.
A group of wards is called a "stake", which is overseen by a stake president and two counselors. All of them must be ordained to the office of a high priest. The stake president and his counselors ("stake presidency") are assisted by a group of twelve high priests called the "high council". As a matter of law I think currently serving members of stake presidencies are recognized as clergy; members of the high council (and former members of stake presidencies, who retain the office of high priest) are not.
, within a ward--if none of the bishopric are present, then the elders' quorum can step in with running the meetings; and if no elders are present, then priests can step in, and then the teachers, and then the deacons. I personally have never seen that happen in a ward, though.
Clear as mud?