Short explanation: Redshift is what happens
to light emitted from a moving body. Cosmological redshift
is a way of
describing the distance to — or, equivalently, the age of —
distant objects.
When an object emitting light moves toward us, the wavelength of the light
will be shortened, making the light appear more bluish. Conversely, if the
source moves away from us, the wavelength will get longer and it will look
more red. In the first case, the light is said to be blueshifted while
in the second case it is said to be redshifted. In both cases, the
effect is caused by the Doppler effect.
An analogous effect is experienced when the sirene of an ambulance moving
toward (away from) us sound more high-(low-)pitched.
From lab experiments we know which wavelengths different elements emit.
By measuring how much the light from a distant object is shifted, it is
possible very accurately to calculate how fast it moves.
In that way Edwin Hubble found out in 1929 that the farther away a galaxy
is from us, the faster it moves away from us, leading to the conclusion that
the Universe expands.
Since velocities away from us, and hence the redshift, depends on the
distance, astronomers often speak of an objects (cosmological) redshift
instead of it distance.
And since light moves with a finite speed (300,000 km/s, actually), we look
further and further back in time, the larger its redshift. That's why the
term is also used as a label of age of the Universe.
