The first, homogeneity, assumes that on large scales, matter is spread uniformly through space. The second, isotropy, assumes that (again on large scales), things look the same in all directions. Finally, universality assumes that the laws of physics are the same everywhere in the universe.
The observable universe is just that part of the universe that is close enough to us that we can receive light from it since the beginning of the universe. In other words, if the universe has age T0, then we will only have received light from objects that are within T0 light years from us. We will see the same distance in all directions as long as there is no edge to the universe. Therefore we are at the center of the observable universe. The universe itself is believed to be much larger than the 10 or 20 billion light years we can see (we believe the universe is between 10 and 20 billion years old). The real universe has no center, but we are at the center of the part that we can see.
A closed universe is finite, but has no edge. The analogy of a two-dimensional ant on a basketball is excellent here. The ant cannot detect an edge, and there is no location the ant can move to that places the ant at the center of its universe (ie the basketball). Nonetheless, the surface area of the universe/basketball is finite. A close (positively curved) three-dimensional universe has the same property.
The expansion of the universe is slowing down as a result of the mutual gravitational attraction of all the matter in the universe. If this matter is sufficiently dense, the expansion will eventually stop, and possibly the universe will begin to contract. If the density of matter is not high enough, the universe will expand forever. This is analogous to another situation: consider a rocket leaving a planet with a particular velocity. If the mass of the planet is high enough, the rocket will eventually fall back to the planet. If the mass of the planet is sufficiently small, the rocket will have enough velocity to escape the planet's gravitational attraction.
In this case the universe is open, and space has negative curvature (a 2-D analogy is the surface of a saddle). This universe is infinite in extent, and will continue to expand and cool forever. The universe eventually becomes cold, dark, and contains nothing but leptons (electrons, neutrinos, etc.).
If the average density of the universe exceeds the critical density, then the universe is said to be closed, and space has positive curvature (a 2-D analogy is the surface of a sphere). Such a universe is finite in extent, and also in time! The universe will eventually collapse back in on itself in a ``Big Crunch''.
The temperature of the hot gases in the universe was about 3000 K at the time of this event. Why then do astronomers say the radiation from the Big Bang has a characteristic temperature of 2.7 K?
We say that the gases of the big bang have a temperature of 2.7 K because these gases have a black body spectrum associated with a temperature of 2.7 K. The gas temperature at the time of the emission of these photons (when the electrons and nuclei combined to form atoms, making the universe transparent to these photons), was around 3000 K. The photons lost energy as their wavelength increased along with the expansion of the universe. The redshift of about 1000 means we detect this radiation at a wavelength that corresponds to the much cooler temperature of 2.7 K.
The Hubble law is the best evidence that the universe is expanding. The recession velocity of an object would be proportional to the object's distance in an expanding universe. Tracing the motions of this expansion back in time leads naturally to a point in time when all matter was in a very small volume, and then the universe began to expand.
The occurrence of the big bang is supported by observations of the cosmic background radiation. Details of this radiation are contained in the answers to the next question.
Finally, the big bang theory predicts that (within the first 30 minutes) the universe was composed of approximately 75% hydrogen and 25% helium (by mass), with trace amounts of a few other elements. Observations of Jupiter, stars, and the interstellar medium show relative abundances very similar to these (accounting for the fact that heavier elements have been formed in the centers of stars).
The Hubble constant is the rate at which the universe is expanding. This is the slope of the velocity versus distance graph (Hubble's law), and has units of velocity divided by distance, or inverse time.
By running the expansion back in time at its current rate, we can estimate the age of the universe. Knowing the speed of travel and the distance between Berkeley and San Francisco allows us to figure out how long the trip will take (or has taken): time = distance/velocity. The Hubble constant has units of velocity/distance=1/time, so 1/H0 is an estimate of the age of the universe (or how long the expansion has taken).
This is an upper limit to the estimate, because gravity (due to the mass in the universe), will slow the expansion, and the universe will not be as old as our estimate. It was expanding faster early on, so didn't take as long to reach its present state. (In our analogy, suppose you left Berkeley travelling at 75 mph, but arrived in San Francisco travelling only 50 mph. Any estimate of the travel time made using 50 mph as your speed will be too long -- the fast speed at the beginning shortened your travel time).
The universe must have been very uniform during its first million years because the cosmic background radiation is very isotropic (ie uniform in all directions, once we remove the effects of our motion relative to this radiation). It is uniform to about 1 part in 100,000! This suggests that the gas at the time that this radiation was emittied was very uniformly distributed.
The biggest problem with structure in the early universe is that we are not sure how it formed. We know that there must have been some structure or the universe would not be as clumpy as we see it today. The presence of quasars early on implies that significant structure had to have occurred in the first 7% of the universe's age. The cosmic background radiation shows minute structure (about 1 part in 100,000). We do not know the cause of this initial structure. The relative uniformity of the cosmic background radiation makes it difficult to achieve very clumpy structure by the time of the quasars. It is believed that initial minute clumps of matter (probably dark matter) grew by attracting other matter through their gravity, but we do not understand the details.