So what's the organization of the church? Like you've already mentioned deacons and elders, and I know that those mean different things in the LDS Church than they do in other churches.
Bladerunner pretty much already got it, but I'll expand on what he already said.
Firstly: deacons, teachers, priests, and bishops are all offices in the Aaronic Priesthood. Elders, high priests, and apostles(of which there are only 70 anywhere) are all offices in the Melchizedek Priesthood. In addition to specific requirements, mostly pertaining to age, for each office, you must have been baptized in order to hold an office in the Aaronic Priesthood, and endowed to hold an office in the Melchizedek Priesthood. The Melchizedek and Aaronic Priesthoods are only for men: the rough equivalent for women would be the Womens' Relief Society, I think. The Relief Society doesn't really have any organized ranks as far as I know, they have a ward president, I don't even know how they are picked. Other than that, they basically just divvy up duties and responsibilities based on how much they think each person can handle. While the Melchizedek and Aaronic Priesthoods handle the clerical duties of the church, the Relief Society is more of a humanitarian thing. They do teach Sunday School, home teaching, but they also do a lot of philanthropic stuff. But I'm not exactly an authority on the Womens' Relief Society.
Deacon
Requirements: 12 years of age.
Responsibilities: Pass out the Sacrament; collect tithings and fast offerings.
Teacher
Requirements: 14 years of age.
Responsibilities: Prepare the Sacrament; go home teaching, usually with their father.
Priest
Requirements: 16 years of age.
Responsibilities: Bless the Sacrament; ordain others to the offices of the Aaronic Priesthood.
Bishop
Requirements: Married adult; high priest of the Melchizedek Priesthood.
Responsibilities: Leader of the ward, preside over and organize Sacrament meetings, judge for temple recommends, oversee financial records from tithings and donations, serve as high priest for the ward, manage the other organizations in the ward(Sunday School, Relief Society, Primary, etc.).
Elder
Requirements: 18 years of age.
Responsibilities: Baptize and lay on hands for others; bless and administer to the sick; teach others; watch over the church. In addition, Elders are the people who go on missions, although that's technically not one of their official duties.
High Priest
Requirements: Responsible adult.
Responsibilities: The responsibilities of high priests depend on what they have been assigned. It's less of an office in and of itself, and a qualification for other offices.
Other offices that you need to be a High Priest to hold are mission president, stake president, bishops, etc.