JOEL - The Church organization is based upon what the Savior wants it to be right now. It was by revelation to Joseph Smith that God directed the Church to be organized in the latter-days with a First Presidency and Quorum of Twelve Apostles, as given in D&C 107:21-24:

21 Of necessity there are presidents, or presiding officers growing out of, or appointed of or from among those who are ordained to the several offices in these two priesthoods.
22 Of the Melchizedek Priesthood, three Presiding High Priests, chosen by the body, appointed and ordained to that office, and cupheld by the confidence, faith, and prayer of the church, form a quorum of the Presidency of the Church.
23 The twelve traveling councilors are called to be the Twelve Apostles, or special witnesses of the name of Christ in all the world—thus differing from other officers in the church in the duties of their calling.
24 And they form a quorum, equal in authority and power to the three presidents previously mentioned.

God said it was "of necessity" to have such presiding officers in order to govern the church in the best way for the times we live in now. The number of councilors is not set in stone. For example President Kimball had three councilors when he was president. Interesting to note that the twelve are "equal in authority and power" to the First Presidency, but they are to be "traveling councilors" to witness to all the world, while the First Presidency takes care of business at Church headquarters.
The First Presidency today represents Peter, James and John, who could be considered the first presidency of the early church; with Peter as the president. These three functioned much the same way as our latter-day presidency does now. For example there were moments in the New Testament when Jesus spoke only with Peter (Matt. 18:19; Luke 24:34), and other times when He spoke to the three of them(Matt. 17:1-3; 26:37-39; Mark 5:37-42).
Also, conditions are a little different today, where we don't actually have Christ physically on the earth ministering to us. If He were here in person, we might not even have a First Presidency; only the Quorum of Twelve Apostles. Christ organizes His Church on the earth for the bennefit of the people who are living here at any given period of time. The most important things we need are, the gospel of Jesus Christ, revelation, and the proper authority to perform the necessary saving ordinances.
The organization is only a tool which is used to administer these other things and can and has been changed to fit the needs of the members of the church at any given time and place.
When we talk with others about the church we are often eager to explain to them that the church is organized exactly the same as when Christ was here on the earth. But that is not completely accurate. It is closer to what He first established than any other existing church today. But more importantly, it does have the exact same gospel He taught and the exact same priesthood authority necessary to administer the ordinances and govern the church. This is what we should emphasize.