|Salt Lake Temple (original photo)|
Two sides of the temple
The Salt Lake Temple has two main sides - the east side and the west side. Each end has 3 towers. The west side represents the Aaronic Priesthood and the east side represents the Melchizedek Priesthood. This two side pattern started with the Kirtland Temple which had 2 large rooms (one on the first floor and one on the second floor). These rooms had pulpits at opposite ends that were designated for the Aaronic Priesthood on one end and the Melchizedek Priesthood on the opposite end. The Salt Lake Temple started the practice of showing this two-ended priesthood symbolism on the exterior. This pattern was copied for the Logan Utah and Manti Utah Temples, was brought back for the Washington D.C. Temple, and revived again for the 1980s six spire temples starting with the Boise Idaho Temple and ending with the Las Vegas Nevada Temple. The San Diego California Temple is a unique version of a two ended temple with its two great towers. Recently, two ended temples have returned starting with the Kansas City Missouri Temple, Brigham City Utah Temple, Rome Italy Temple, Philadelphia Pennsylvania Temple, Fortaleza Brazil Temple and potentially with other temples in planning.
Three towers on each end of the temple
The west and east ends of the Salt Lake Temple each have 3 main towers. These are used to represent the priesthood leadership. The 3 Melchizedek Priesthood towers on the east end represent the First Presidency or a stake presidency - the leadership of the Melchizedek Priesthood. On the east side the three Aaronic Priesthood towers represent the Presiding Bishopric or a local bishopric - the leadership of the Aaronic Priesthood. Some other temples have kept the 6 tower symbolism. The Logan and Manti Utah Temples each have smaller side towers that are often overlooked. The Washington D.C. Temple and 1980's six spire temples also include 6 total towers. The Brigham City Utah Temple has also included smaller side towers to keep the 6 tower symbolism intact. A lot of the other two ended temples lack the 6 towers symbolism.
You might notice little spires on the towers of the Salt Lake Temple. On each tower of the temple there are 3 levels of 4 pinnacles. This makes 12 pinnacles on each tower (in addition to the main point of the tower). The pinnacles on the east end represent the 12 apostles. The Bountiful Utah and Mount Timpanogos Utah Temples both have 12 circular windows at the top to represent the same thing (6 go into the celestial room and 6 into the chapel). I have read that the 12 pinnacles on the west end of the Salt Lake Temple represent the high council, although I am unable to track down this explanation and am unsure how this relates the the Aaronic Priesthood.
Different Tower Heights
On the Salt Lake Temple, the east towers are 6 feet taller than the west towers. This is to represent the Melchizedek Priesthood being above the Aaronic Priesthood. This symbolism has also been done in the Logan Temple, Manti Temple, and many others. The center towers on each side of the Salt Lake Temple are also taller than their side towers representing the President of the Church, stake president, Presiding Bishop, or bishop leading among their counselors. The Washington D.C. Temple takes this symbolism even further by having all 6 towers at different heights which would show the relative position of a first counselor and a second counselor.
The windows on the Salt Lake Temple towers also contain symbolism. The western towers have 4 levels of windows and the eastern towers have 5 levels of windows. These represent the offices in the priesthoods. The four Aaronic Priesthood offices are deacon, teacher, priest, and bishop. The 5 Melchizedek Priesthood offices are elder, high priest, patriarch, seventy, and apostle. Windows are a fitting symbol as they bring in light, as does priesthood and the revelations associated with it. This window symbolism developed gradually - earlier temples lacked it and early plans for the Salt Lake Temple didn't include it.
So that is some of the priesthood symbolism in the Salt Lake Temple Towers. Priesthood symbolism is important because the temple is very much about the priesthood and the 2 priesthoods are important to the ordinances of the temple.
Please comment with any insights you may have on this topic.
Here are some of my references:
https://www.lds.org/manual/doctrine-and-covenants-and-church-history-gospel-doctrine-teachers-manual/lesson-8-the-restoration-of-the-priesthood?lang=eng See additional teaching idea 1.