Freeman Dyson noted that one of the limiting resources for
civilizations is the amount of energy they can harness.  He proposed
that an advanced civilization could harness a substantial fraction of
its sun's energy by enclosing the star in a shell which would capture
most of the radiation emitted by the star.  That energy could then be
used to do work.

As originally proposed a Dyson sphere consisted of many solar
collectors in independent orbits.  Many science fiction writers have
modified the idea to make a Dyson sphere one complete shell.  In
addition to capturing all of the available energy from the star, such
a shell would have a huge surface area for living space.  While
Dyson's original proposal of a number of solar collectors is stable,
this later idea of a complete shell is not stable.  Without some
stablizing mechanism, even small forces, e.g., a meteor hit, would
cause the shell to drift and eventually hit the star.  Also, the
stresses on a complete shell Dyson sphere are huge and no known
material has enough strength to be used in the construction of such a
shell.

There have been searches for Dyson spheres.  Such searches typically
take place in the infrared.  Because the shell is trapping energy from
the star, it will begin to heat up.  At some point it will radiate as
much energy as it receives from the star.  For a Dyson sphere with a
radius about the radius of Earth's orbit, most of the radiation
emitted by the shell should be in the infrared.  Thus far, no search
has been successful.

Considerably more discussion of Dyson spheres is in the Dyson sphere
FAQ, <URL:http://www.student.nada.kth.se/~nv91-asa/dysonFAQ.html>.