A Dyson Sphere is a theoretical mega-structure constructed to collect much or even all of the solar radiation produced by a star, extract energy via solar panels or some similar technology, and then radiate lower-energy infrared radiation out into space. It was first described by Freeman Dyson as a mechanism for a sufficiently advanced civilization, either human or alien, which had grown beyond the energy production capabilities of a single planet.
Dyson, a physicist, devised the sphere as a hypothetical solution to the problem of indefinitely increasing energy requirements. Since the invention of stone tools and then of farming, he observed, every successive civilization had been dependent upon greater energy production than those which came before. (Today, similarly, we rely on much more power from electricity, petroleum, and nuclear fission than all previous eras in human history.) Assuming that technological progress and therefore energy demands continued to increase indefinitely in the future, it followed that humanity would sooner or later reach a point where we were no longer capable of satisfying our energy demands using only the production capacity of a single planet like Earth.
Furthermore, Dyson observed, there was reason to believe that intelligent life on other planets, elsewhere in the galaxy and the universe, would, given sufficient inspiration and natural resources, follow a similar ladder of increasing energy consumption as we have on Earth. They, too, would eventually turn to energy sources off their native planet. If these alien species were older than humanity, then they might already have done so.
The next step after harnessing the energy capacity of a planet would be to harness the capacity of the Sun. Dyson theorized that a sufficiently advanced intelligent civilization, potentially even future humans, could launch a vast network of solar power collection satellites into stellar or solar orbit, forming a ring around its sun. This approach is now known as a Dyson Swarm. A much more ambitious and powerful solar power collection system, known as the Dyson Sphere, would see an incredibly advanced civilization construct an entire shell around their star, perhaps at the distance of Earth's orbit or more, so that the entire solar output of the star could be harnessed.
For obvious reasons, a Dyson Swarm is a much more plausible construct than a Dyson Sphere. Not only are the material requirements far less breathtaking in scope, but the swarm could be constructed incrementally through the launch of successive waves of solar power satellites, rather than requiring a single mammoth construction project. Moreover, the construction of even a very thin shell around a star at any substantial distance would presumably require the conversion of enormous amounts of material for the purpose, perhaps even the deconstruction and harvesting of entire planets for raw materials. Obviously anything on this scale is well beyond what humans can conceive of today.
In addition to the material requirements, there is another theoretical problem with a Dyson Sphere that might make it more science fiction than science fact. First, a sphere overlaying a star could not be held in place by the gravitational influence of the star, unlike an object in orbit. This means that it would either require a separate and even more unfathomably powerful system of propulsion for stationkeeping purposes, or that it would be in constant danger of drifting out of position and being destroyed.
Finally, constructing a shell around a star would have immense consequences for the broader dynamics of the solar system. Although stellar gravity would still operate, the effects of the solar wind, like the maintenance of the heliosphere which protects us from the matter in the interstellar medium, would probably dissipate. It is uncertain whether life actually benefits from the heliosphere - and it will be difficult to determine otherwise until we send space probes outside of the solar system to conduct such tests. However, it is important to remember that the stream of radiation and particles emitted by the Sun would be stopped against the inner layer of the shell of the Dyson Sphere - so that anything outside of the sphere would be rendered darker and colder than before. In our own solar system, we would essentially be spelling the end of any future life that might plausibly exist on Europa or Ganymede, as well as rendering the rest of the system a very cold, dark place indeed.
It should be noted that all of these are problems of scale, rather than of theory. We could construct the first elements of a Dyson Swarm today simply by launching solar panels into solar orbit (although connecting these and drawing power from them would be a more troublesome task). Moreover, a Dyson Sphere would not violate any current theories of physics, provided that sufficient building material could be found. A propulsion system for stationkeeping purposes could perhaps be devised, and some compensation for the loss of the heliosphere could perhaps be projected. However, these are not minor challenges. The actual construction of a Dyson Sphere would require technology so far in advance of our own that the sphere itself is a comparatively minor phenomenon in comparison.