What is meant when it is said that the universe is homogeneous and isotropic?
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.
According to the generally accepted theory of the Big Bang, the universe originated between 10,000 and 20,000 million ago years ago and has been expanding ever since.
The uncertain future of the universe: the expansion could be limited (closed universe), shrinking the universe upon itself, or it could be infinite (open universe), in which case the universe will expand forever.
In the limiting case between these two possibilities (flat universe) nor cease the expansion. At large distances from the observational point of view, the universe is - homogeneous - isotropic.
If $u$ is the mass density, call $(du / u)$ in $R$ a fractional fluctuations of the mass density, due to the existence of structures (galaxies, galaxy clusters, superclusters), within an area located randomly, radius R. Thus $(du / u)$ in $R$ decreases as a negative power of R.
This suggests that we can model the universe as made of a soft background of average density, with superimposed fluctuations. A small scales these fluctuations are large but decrease as the scale grows. A sufficiently large scale (greater than about 100 Mpc), the universe can be treated as homogeneous and isotropic then a uniform density.
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