General - Stellar Cosmology
A world is defined as much by the space around it as it is by its surface. Earth's geography, climate, and society are significantly affected by the fact that the Earth has one sun and one moon. Tamriel, the world of The Elder Scrolls game franchise, has a moon, but the sun and stars are actually holes in the native plane caused by the departure of a demonic deity and his followers tearing holes in the border of the plane. Tatooine, the iconic desert world of Star Wars, has two suns but no moon. Eberron has a twelve small moons, but also has a ring made up of magical crystals which occasionally rain upon the world below, and are used for all sorts of religious, magical, and decorative purposes.
In the real world, most planets revolve around stars. A planet large enough to be orbited by a star is physically impossible: anything with enough gravity to outweigh a star would either become a star or a black hole. In your world, all bets are off. The sun may not be a sun at all; it might actually be a divine being flying around the earth on a chariot drawn by winged horses (looking at you there, Helios), or it may be the eye of some huge cosmic being gazing down upon the world below. Or, it might be completely mundane and the people of the world might lack the capacity to understand the mechanics of such a cosmic body, and invent some grand myth to explain it (again, Helios).
Science Fiction and Science Fantasy settings which involve space travel may want to devote more detail to the exact function of the world's solar body, and you may go so far as to pick the class(es) of star(s) (white dwarf, etc.), assuming of course that the solar body is in fact a star.
Moons have considerable effects on the world they orbit. The moon's gravitational pull affects the Earth's tides, and the light provided by the moon has plenty of effects which are beyond my limited capacity to explain. Moons are useful for measuring time and the passage of the seasons, and have featured heavily in mythology throughout history. Moons are numerous and varied, even in our own solar system, so feel free to get creative with your world's lunar bodies.
Stars and Other Distant Astral Bodies
The stars visible in the Earth's night sky have held divine, philosophical, and scientific significance for as long as man has gazed at the stars. Every society has some sort of idea about the significance of the lights in the night sky, just as they have some belief about the sun and moon. Stars in your world might be the same sort of stars we see in the real world, or they might even be those same stars if your setting is set in a real-world analogue. They might be lights on the ceiling of some unimaginably large dome encircling the world, or they might distant spirits looking down on the living far below.
Example 1 - Shadow of Olympus
The Greeks had a very interesting concept of the cosmos which I'm going to outright steal. The world is flat, and the sun rises from an actual place on the world, and sets when Helios lands on the opposite edge of the world. The stars were placed there by the gods, and are generally composed of creatures and people to whom something interesting happened. The moon is personified by Selene, and moves around the sky something akin to how the flat earth theorists believe it does. There are no other planets, though occasionally moving stars move across the night sky as the gods go about their business. As discussed in our planar cosmology, the material world is a roughly hemispherical dome with Hell hanging off of the bottom side, and the elemental chaos knocking against the glass, looking for cracks.
Example 2 - Space Grease
I want Space Grease to have a vaguely familiar feel, so we can use Earth's solar system. I imagine a lot of the action taking place around the outer planets like Jupiter and Saturn.
Example 3 - Heroes of Tonesvale
Since our heroes setting takes place in a fictional city in the real world, it's perfectly fine for us to use Earth's solar system.