The Discworld is lit by a tiny 'sunlet' which orbits the disc at a rate of once per day, but is this miniature sun orbiting the disc itself, or around Great A'Tuin?
In the novels it's not entirely apparent how the sun (and moon) work in terms of celestial mechanics. In earlier works its orbit is described as being 'complicated' and although the strong inference is that while the sun does periodically travel under the disc (e.g. between A'Tuin and the Discworld), it also seems to periodically orbit the World Turtle as well.
Through the fathomless deeps of space swims the star turtle Great A’Tuin, bearing on its back the four giant elephants who carry on their shoulders the mass of the Discworld. A tiny sun and moon spin around them, on a complicated orbit to induce seasons, so probably nowhere else in the multiverse is it sometimes necessary for an elephant to cock a leg to allow the sun to go past.
It was midnight on the Disc and so, therefore, the sun was far, far below, swinging slowly under Great A’Tuin’s vast and frosty plastron. Rincewind tried a last attempt to fix his gaze on the tips of his boots, which were protruding over the rim of the rock, but the sheer drop wrenched it away.
The Colour of Magic
Expletius had proved that the Disc was ten thousand miles across. Febrius, who’d stationed slaves with quick reactions and carrying voices all across the country at dawn, had proved that light travelled at about the same speed as sound. And Didactylos had reasoned that, in that case, in order to pass between the elephants, the sun had to travel at least thirty five thousand miles in its orbit every day, or, to put it another way, twice as fast as its own light.
It [the Discworld] is lit by one small sun which moves in a very complicated orbit. As a result, this is probably the only place in the universe where is is sometimes necessary for an elephant to cock its let to let the sun go past
And the intro for the show has the sunlet traveling under A'Tuin, and between the world turtle and The Disc in the outro.
Interestingly, this is all flatly contradicted in the Ultimate Discworld Companion which states that the orbit is stable(!) and travels between the turtle and the disc, albeit at an oblique angle so that it only passes between two of the elephants rather than between all four.
The tiny sun orbits in a fairly flat ellipse, being rather closer to the surface of the Disc at the Rim than at the Hub (thus making the Hub rather cooler than the rim). This ellipse is stable and stationary with respect to the Turtle – the sun passes between two of the elephants.
The Ultimate Discworld Companion
I hate to say it but, I'm pretty sure Discworld changes at the whim (or perhaps forgetfulness) of the author.
In one of the early books, the year and seasons get spelled out. The year is twice as long as ours, with two winters and two summers each. Mid-summer happens when sunrise or sunset (correct terms for the Disc!) happens nearest to where you are. Winter, when you are far from that. They had the terms "Fimbulwinter" (sp?) and "Spinwinter". I believe Hogswatch was mid-Fimbulwinter. All of this means that if you move half a degree around the Disc, your seasons shift by a day. Hogswatch was determined by where you are. I believe this also made seasons more pronounced the farther from the hub you were.
In a later book (Hogfather, I think), the year reverted to what we have. With no particular explanation of how you got seasons in the first place.
So, net, the orbit of the sun probably changed in different stories, for no particularly good reason.