The sun is the powerhouse for our solar system, heating the spheres with its radiation and holding everything from gas giants to asteroids in thrall to its massive gravity. Stars, in their various forms, are the most powerful forces in the known universe, so it only comes naturally that a species of egotistical apes would dream about capturing one and bending it to their will.
I am of course talking about us (not these guys), the people who can't look at the moon or a hurricane without fantasizing about nuking it. Granted, these ideas are largely what-if experiments, but they've certainly taken hold in science fiction. And if we thrive long enough to became a Type II civilization on the Kardashev scale*, well, here are some of the things we might do with our pet stars:
Alderson disk: Imagine a giant CD with an entire star situated in the middle of it. That's an Alderson disk, named for U.S. scientist Dan Alderson. The disk would be so huge that we'd have to strip whole planets for the necessary materials, and the resulting mass would provide the disk with its own gravity (this holds for all the massive structures in this post). As such, each side of the disk could contain whole civilizations, though only toward the middle of the disk, where it's neither too cold nor too hot. Enjoy the perpetual twilight and lack of seasons, Goldilocks.
Ringworld: If you're a fan of science-fiction books or, failing that, video games in which you shoot things, then you're probably familiar with this concept. It's very similar to the Alderson disk, only instead of using a sun-dwarfing CD, it uses what amounts to a giant bicycle tire which rotates around a central sun. Also known as a Niven Ring, for "Ringworld" author Larry Niven, the inside of the great wheel (facing the sun) would contain spreads of Earth-like atmosphere, geography and cities. Niven's vision also entailed an inner ring of alternating shade panels to provide periods of "night." The downside? No romantic sunsets, possibly disastrous imperfect physics and, if the novels are any indications, possible cat people invasions.
Dyson Sphere: If you want to be really stingy with your pet star, then this is the way to do it. British theoretical physicist Freeman Dyson dreamed up the concept, in which the sun is encircled by clouds of orbiting, enclosed habitats and solar collectors. This would allow the builders to better harness the energy of the star. Such constructions could, in theory, eventually reach the point where they completely enclose the sun like a shell. This is the concept you encounter in such science fiction as "Star Trek: The Next Generation" and Warhammer 40K. According to David Darling's excellent write-up, only the equatorial zones would be inhabitable and even the slightest bump from an asteroid might cause a collision of star and the structure. For Dyson's part, he dreamed up the concept as something we might look for (in the form of an expelled inferred waste ration) when scanning the galaxy for signs of highly advanced civilizations. So heads up alien civilizations, you can't hide from us forever in your massive spheres.
Shkadov thruster: While the above concepts all involve harnessing the power of a sun to power both artificial ecosystems and technology, the Shkadov thruster takes things one step forward: turning the whole structure into a starship powered on its own captive sun. The concept, envisioned by Leonid Mikhailovich Shkadov, involves the use of mirrors to redirect a star's energy. His theory, according to Astronomy Today, was to place solar mirrors near the sun and redirect the radiation so as to push the star and all its gravity-bound objects in a direction of our choosing (see How Solar Sails Work). In effect, our entire solar system becomes a space ship we can navigate. There are three classes of stellar engines, explained in more depth here and another variant, called a Matrioshka Brain, which harnesses a star's energy entirely for computer processing needs.
So there you have it: four ways to exploit a star that don't involve drugging a celebrity and walking him or her to an ATM machine. Use with caution, type II readers.
* According to Nikolai Kardashe's speculative 1964 scale, a type I civilization can harness the net power of a single planet, a type II can harness that of a single star and a type III has the almost unimaginable power to harness the power of an entire galaxy.