Expanding Universe
The expansion of the Universe is not an expansion of space into another surrounding space, but it is the expansion of space itself. This is difficult to understand because we do not have any phenomenon in our daily life to compare this with.
Ants on an inflating balloon or the Expanding Universe One way to imagine what it means is to watch an inflating balloon. The surface of the balloon is a 2dimensional representation of our 3dimensional space. On it the distance between any two points increases when the balloon inflates and points that are twice as far apart move twice as fast. Note that the ants do not move themselves on the balloon! 
For a similar demonstration watch this video but with this warning.
The expansion of space is now a fundamental part of the Cosmological Models that are used to describe the evolution of the Universe. As a consequence, the distance between any two objects in space (e.g. galaxies) is increasing. The further these objects are apart, the faster they will recede from each other. Hubble’s Law describes this expansion. The apparent motion due to the expansion is called the Hubble Flow. It does not mean that galaxies are actually moving with respect to each other.
But if we measure redshift of objects that are very far away, what do we actually measure?
Looking at the simple formula for Optical Redshift we used before (the actual Doppler effect)
we see that if the redshift is equal to 1 the velocity of the object is equal to the speed of light. But nowadys we routinely measure redshifts that are much larger than 1.
This means two things:
 When we apply the theory of general relativity (which we must for velocities that are a significant fraction of the speed of light) we see that the expansion at large distances actually is going faster than the speed of light. Since this is an expansion of space itself, it does not violate the speed limit c, dictated by special relativity. For these calculations we need a cosmological theory that also involves the Hubble constant. This is not a Doppler effect.

Cosmogical Redshift means that photons are "stretched" while they have been travelling towards us. This "stretching" is observed as a change towards longer wavelength. The larger the cosmological redshift we observe, the longer the photons have been travelling, i.e. the longer ago they left the source. The actual meaning of the cosmological redshift is that it tells us how much the universe has expanded since the light was emitted.
The wavelength of the emitted radiation is lengthened due to the expansion of the Universe. In this animation, the galaxy on the left was formed a long time ago, while the galaxy on the right was formed more recently. Although each galaxy emits the same wavelength of light, the light from the left hand galaxy has spent longer travelling through the expanding Universe, and has therefore experienced a greater ‘stretching’ (redshift). Thus the more redshift is measured the longer these photons have been travelling. Source: Cosmos, Swinburne Astronomy Online 
But the key question for our discussion is: Can cosmological redshift be a measure for distance?