It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?
A homogeneous cosmology is one in which there are no "special" places in the universe: at a given instant in time, the universe appears the same at every location (on large enough spatial scales).
An isotropic cosmology is one in which there are no "special" directions: at a given instant in time, the universe appears the same in every direction (again, on sufficiently large spatial scales).
Together, they form the Cosmological principle.
As pointed out by Brian Hooper, these symmetries (when applied to physical laws) give rise to the conservations of linear and angular momentum as a result of Noether's theorem.
In addition, the cosmological principal is important for the physical interpretation of observational data, and not only because it is a generally unspoken assumption when using physical laws tested on Earth to model distant objects (galaxies, quasars, etc.) For example, it supports the interpretation of the Hubble diagram as the result of the expansion of the universe as opposed to evidence that the Earth (or someplace "nearby") was at the center of a very big conventional explosion. After all, in a conventional explosion, the fragments that travel the farthest are those that had the highest velocity, so some time after the explosion, faster moving fragments are further away from the center. If, however, observers on all fragments see the same density of galaxies and relationship between velocity and distance in all directions, then this model doesn't work.
It means that the laws of physics are the same everywhere and the same in every direction. It is of fundamental importance as these symmetries give rise to conservation laws. The isotropy of the universe means that angular momentum is conserved; its homogeneity means that momentum is conserved. A similar symmetry, that the laws of physics are the same for all time, gives us conservation of energy.
See Noether's Theorem on Wikipedia for more information.
Most of modern cosmology is based on the Cosmological Principle, which states that the spatial distribution of matter in the Universe is homogeneous and isotropic when viewed at a sufficiently large scale such that there are no observable irregularities. Studies of large-scale structure in the universe and analysis of the microwave background radiation help confirm that this assumption is justified. The validity of the Cosmological Principle relies on the simultaneous homogeneity and isotropy of the Universe. On large enough spatial scales, the notion of homogeneity means that there are no special places in the Universe and at a given instant of time the Universe appears the same at every location, and the notion of isotropy means that there are no special directions at a given instant of time and the universe appears the same in every direction.
In order to link homogeneity and isotropy, we may invoke the "Copernican principle," that we do not live in a special place in the universe. Then it follows that, since the universe appears isotropic around us, it should be isotropic around every point; and a basic theorem of geometry states that isotropy around every point implies homogeneity. However, the opposite is not true. Also, a Universe that is isotropic around only one point is not homogeneous.