A sidereal day is the amount of time it takes for the Earth (or, in general, any
planet) to make precisely one revolution, relative to the Celestial Sphere. Distant
stars make good approximations of fixed points on the celestial sphere, so a
sidereal day can be considered to be the time between a star reaching zenith (or it's
cloest approach to zenith) and it's reaching zenith again.
A solar day is the time it takes for a planet to make precisely one revolution,
relative to the Sun (or whatever star it is orbiting). In other words, this is the
time from noon (sundial time) to noon (sundial time).
It is difficult to imagine the difference between the two without a diagram, so here's
one. Imagine a planet where the (sidereal) year is only eight solar
days (and for simplicity imagine the planet has no axial inclination).
orbit







\
\

\ B
9
A \
 star at zenith for B8
/ A8B             
\
sun at / A
zenith 7
for A9 / B
A
/ 6
B
/  A
5
/ B

/ B4A

/ B
3
/ A
B
/ 2
A
/ B
1
/ A
sun at zenith for A0  star at zenith for B0
*                  A0B             
Imagine two observers on opposite sides of the planet at some time 0, so that it's noon
for observer A (and consequently midnight for observer B). As time passes, the planet
spins and goes around in it's orbit. (In this diagram, the spin and the orbit are both
in the same direction  anticlockwise. The Earth's orbit and spin looks the same when
viewed from some point to the North of the ecliptic, except of course it spins much
faster.) At time 8, the planet has made one complete revolution (relative to the celestial
sphere), so the same star that was at zenith for B at time 0 is at zenith once more. The
sidereal day is 8 units long. However, as observed from the planet, the position of the
sun has changed relative to the celestial sphere, so observer B doesn't see the sun at
zenith until time 9. The solar day is 1/8 longer than the sidereal day.
It is no coincidence that 8 is the number of solar days in the year, and 1/8 is the
factor by which the solar day differs from the sidereal day. Over the course of a year,
the number of sidereal days is precisely one more than the number of solar days. Thus,
there are about 365.25 solar days in the year, but there are about 366.25 sidereal days.
This difference of 1 can be conceptually grasped by considering a planet that is in tidal lock with the sun  imagine observer A always seeing the sun at zenith in the diagram above, and
observer B always on the opposite side. There are no solar days at all, but when the planet
makes one revolution in one year, observer B sees the same star at zenith as a year before 
so there's one sidereal day where there are no solar days. (If the planet were spinning in
the direction opposite the orbit, with more than 1 solar day per year, there would be one
fewer sidereal days than solar days. If the planet were spinning in the opposite
direction, with less than 1 solar day per year, the sum of the fraction of the solar day
per year and the fraction of the sidereal day per year would be 1.)
Earthlings most easily recognise the solar day because the sun is so significant in the
sky; it lights up the whole sky by atmospheric scattering and obscures all the other
stars, so the solar day is the most significant type of day on Earth. On Neptune or
Pluto, however, the sun (at first glance) looks like it's not much more than just another
star  brighter than usual, but it does not have such a glare as to make all the other
stars invisible. On distant, airless planets, the sidereal day would be more recognisable
than the solar day. Neptunians would be more likely to make their calendars, and calculate
their dates and times of 'day' by sidereal time rather than solar time.