It is almost totally improbable for a toroidal star to originate naturally, but in theory it may be possible for a very technically advanced society to create one artificially, probably involving a series of massive linear accelerators.
If a large body of gas was spun up to sufficiently high rates of angular velocity, such that the centripetal forces were greater than the force of gravity at the surface of a sphere of the amount of gas involved, then it is possible that the gas could form a toroid.
With a sufficient volume of gas, compression within the toroid could lead to the ignition of a hydrogen fusion reaction, resulting in the toroid being the requested luminous band.
The amount of gas involved would be extreme - without the high-velocity spin keeping the ring-star a ring, the volume of gas involved would form a hypergiant star or a Wolf-Rayet star, which would quickly shed a lot of its mass. The speed of rotation would also be extreme, probably on the order of thousands of kilometres per second, depending on the diameter of the torus.
In order to have a world at the centre of such a system, the world could simply exist at the centre of mass/gravity of the system, and be constantly lit at the equator, with twilight at the poles.
Alternatively, we could invert another proposed type of solar system, where there is a central sun that bobs through the hole in a large solid flat disk (an Alderson disk) - in this system, a central world could bob through the centre of mass of the ring star, thus alternately lighting first one side of the planet then the other, providing a day-night cycle.
As an aside, it may be very difficult to explore such a system without very powerful sources of thrust - the toroidal star would have a massive gravitic field, and would play havoc with conventional rockets' trajectories, meaning that it would be very difficult to visit bodies in the outer system, if there were any, and even seeing them from a planet at the centre of the ring would be difficult if they were in the plane of the toroidal star's rotation.