While on the face of it, this seems a reasonably easy question with a
simple answer, like the "When does the 21st Century begin?" question
there is no hard and fast rule, no committee of astronomers who decide
these things.  The best rule of thumb is that if people think
something's a planet, it is.  Common criteria include orbiting the Sun
rather than another body (although sticklers find this troublesome)
and being "large".  Some have suggested using "world" as a neutral
term for an interesting solar system body.  The word "planet"
originally meant "wanderer", so using a strict definition, everything
in the solar system is a planet!

When Pluto was discovered in 1930, there was no question as to whether
it was a planet.  The predictions made at the time imagined it to be
at least the size of the Earth.  As better data became available with
the discovery of Pluto's moon Charon allowing the determination of a
mass for Pluto, and with Pluto and Charon eclipsing each other in the
late 1980's--early 1990's, it was found that Pluto is much smaller
than the Earth, with a diameter of roughly 2300 km (or about 1400
mi.).  In the last several years, a number of small bodies at about
the same distance from the Sun as Pluto have been discovered,
prompting some to call Pluto the "King of the Kuiper Belt" (the Kuiper
Belt is a postulated population of comets beyond Neptune's orbit) and
rally for its demotion from bona-fide planet to overgrown comet.

Is Pluto a planet?  It depends on what one thinks is necessary to
bestow planetary status.  Pluto has an atmosphere and a satellite.  Of
course, Titan has a much larger atmosphere, and the tiny asteroid Ida
has a satellite.  Most astronomers would probably consider stripping
Pluto of its status akin to stripping [the U.S. states of] Connecticut
or Vermont of statehood because Texas and Alaska later joined.

Is Ceres a planet?  Like Pluto, when it was first discovered there was
no doubt that it was.  Within a few years, however, Pallas, Vesta and
Juno were discovered.  While Ceres is the largest asteroid, the
second, third and fourth largest asteroids are roughly half its size,
compared to Pluto, which is about ten times larger than the Kuiper
Belt objects found so far.  Ceres is also not thought to have
undergone large-scale geological processes such as vulcanism, although
Vesta has.  The consensus is probably that neither Ceres nor any other
asteroid is a "planet", though they are interesting bodies in their
own right.

Is Titan a planet?  In the 1940's a methane atmosphere was discovered
around Titan, making it the only satellite with a substantial
atmosphere.  This atmosphere has long prevented observations of the
surface, frustrating the attempts of Voyager 1 and 2 and leading
theorists to suggest a Titan-wide global ocean of carbon compounds.
Recent observations have been able to penetrate to the surface of
Titan, showing tantalizing glimpses of what may be continents on the
surface.  The atmosphere combined with Titan's large size have led
some to consider Titan a "planet", but what about Ganymede, which is
larger, or Mercury which is smaller and has no atmosphere?  Again, the
general consensus is that satellites are not planets.