Cosmology describes the planets thusly:
The planets are the gods and the planes of the gods, which is the same thing. That they appear as spherical heavenly bodies is a visual phenomena caused by mortal mental stress. Since each plane(t) is an infinite mass of infinite size, as yet surrounded by the Void of Oblivion, the mortal eye registers them as bubbles within a space. Planets are magical and impossible. The eight planets correspond to the Eight Divines.
Nirn (Ehnofex for 'Arena') is a finite ball of matter and magic made from all of the god planets at the beginning of time, when Lorkhan tricked/convinced/forced the gods to create the mortal plane. Nirn is the mortal plane and the mortal planet, which is the same thing. Its creation upset the cosmic balance; now all souls (especially the Aedra-Daedra/Gods-Demons) have a vested interest in Nirn (especially its starry heart, Tamriel).
This video shows the Orrery of Stros M'kai as it appeared in TESA: Redguard.
The Elder Scrolls IV: Oblivion released a DLC wherein the player could help reconstruct an orrery in the Arcane University. Structurally, it appears much like the originally orrery in Redguard. A few important differences exist, though. While Redguard had all planets floating in space, Oblivion's orrery instead anchors some planets on orbital and others on large metallic "wings."
Masser and Secunda are shown on orbits around Nirn, offset from each other by a few degrees. Kynareth is also on an orbital, though on a very different angle than either of the moons. Julianos and Akatosh both orbit Nirn, but are not shown with orbitals. Stendar is shown on an orbital around Julianos. Mara is on an orbital around Zenithar, and Dibella on an orbitals round Mara. Zenithar itself rotates about Nirn's axis, but not around Nirn itself. Arkay is a similar case, revolution above Nirn's tilted axis, creating a sort of 8 shape when seen in time lapse.
A graph of the orbits and orbitals as depicted on Oblivion's orrery.
A gif of the planets in their rotations. Unlike above, these have no had their orbits rotated to all be on the same axis, and, as such, many appear elliptical rather than round.
Elder Scrolls Online provides our third example of an in-game orrery. The orrery of Elden Root seems to have more of a ceremonial purpose than a cosmological one - it is used primarily to discern who among the Altmeri stock is fit to rule Tamriel. As such, it is a far simpler design. While the planets on this orrerly are unlabeled, we can surmise that they are likely Kynareth and Akatosh based on their positions.