The observable Universe is about 10 billion light years in radius. That number is obtained by multiplying how old we think the Universe is by the speed of light. The reasoning there is quite straightforward: we can only see out to that distance from which light can have reached us since the Universe began. 

We determine the age of the Universe in a number of ways. One is to estimate the age of the oldest stars we see. Our knowledge of how stars of a given size evolve with time is very good (based on what we know about atomic and nuclear physics) so the major uncertainty here is usually measuring how far away (and so how big) such stars are. The standard method is to look for very small changes in the apparent positions of the stars as the Earth moves around the Sun. (This effect is called parallax). A second way to get an age for the Universe is to try to figure out the time of the big bang itself. Here the method is to use a series of techniques (based on how bright things appear to be - like Cepheid variable stars - that we think we know the true brightness of) to determine first the distance of the nearby galaxies, then increasingly distant galaxies, until we have estimated distances for many galaxies for which relative velocity measurements have been made (using the Doppler red shift of features in their spectra). The relative velocities we observe for distant galaxies have been largely determined by the expansion of the Universe begun with the 'big bang'. So, once we've determined how expansion velocity correlates with distance for some range of distances, it's possible to extrapolate back (with some assumptions) to calculate the instant of the big bang, when all the matter in the Universe was at a single point.

We also observe that massive objects attract each other through the gravitational force. This force tends to contract matter locally (for example, a gas cloud condenses to form a star). On the large scale you can think of the expansion of the Universe acting to separate galaxies from one another, and the gravitational force acting to attract them toward one another.

The end of time depends on just how much mass there is in the Universe. We talk about this in terms of the density of the Universe, and compare densities to the critical density. If the density is greater than the critical density, then eventually gravity will overtake the expansion. The expansion will slow down and eventually reverse, so that the Universe will be contracting. Eventually it will end in a collapse (or a bounce) called the Big Crunch. If the density is less than the critical density then the Universe will continue to expand forever, with the gravitational force never overtaking the expansion. An ongoing area of research is to measure the density of the Universe. Currently, some observations (and some theories) indicate that the density of the Universe is very close to the critical density. In this case the expansion will slow down so that it is approaching zero expansion as time approaches infinity.

It has been said that because our universe creates its own space and time it is expanding into pure nothing. An infinitely dense point of matter appears in an otherwise perfect vacuum state. An event of some sort causes that point to begin to expand very rapidly, overcoming whatever initial gravitation pull would keep the point of mass together. As the new universe continues to expand, there is less and less gravitational pull to bring all of this mass back to its origin. If the pulling force outward is constant and the gravitational pull continues to decrease, the expansion rate will continue to increase. Apart from that potentially damaging argument, there is the issue of the definition of universe. Or universe, by one definition, is everything. There is nothing beyond or outside of it, not even the empty space-time we can conceive of as perfect space, so there would be no vacuum into which the universe could expand. This may seem a bit of a paradox, as we can always imagine something outside of our house or our solar system, but then it really becomes a question of philosophy as much as science.

In a cluster of galaxies, we can estimate the masses of stars in the galaxies and the hot gas that fill the cluster. We can also infer the total mass of the cluster that is needed to keep it gravitationally bound. The latter is typically found to be ~5 times the combined mass of the stars and the hot gas; an analogy with our Galaxy suggest that only some of the dark matter can be MACHOS. Although circumstantial, such results point strongly to the presence of non-baryonic dark matter in the clusters of galaxies. When it comes to deciding what kind of exotic particles may make up the non-baryonic dark matter, however, there may be a hint of weakness, in that different particle physicists favor different exotic particles. Moreover, as far as I know, there has not been a direct detection of these exotic particles.

Two likely possibilities for the dark matter in our own galaxies are MACHOs (MAssive Compact Halo Objects) and WIMPs (Weakly Interacting Massive Particles). MACHOs are low mass stars, brown dwarfs, neutron stars and white dwarfs. If MACHOs make up most of the dark matter, the distribution is not smooth on the scale of the Solar system, but it is smooth on a much larger scale. If the Galactic dark matter consists of WIMPs, then they are dispersed throughout the Galaxy, with a distribution somewhat different from that of the stars that we can see. Since gravitational pulls of WIMPs from different directions tend to cancel out, the orbit of planets in our solar system is not affected by the presence of WIMPs. However, since there are more WIMPs towards the centre of our Galaxy than away from it, the motion of the Solar system (and other stars) in the Galaxy is strongly affected. This is how astrophysicists infer the presence of the dark matter.

1) It must orbit the sun, 

2) be massive enough that its own gravity pulls it into a nearly round shape, and 

3) be dominant enough to clear away objects in its neighbourhood. 

To be admitted to the dwarf planet category an object must have only two of those traits -- it must orbit the sun and have a nearly round shape. And no, moons don't count as dwarf planets. 

In addition to Pluto, Ceres and 2003 UB313 the astronomical union has a dozen potential 'dwarf planets' on its watchlist. What's to become of the other objects in our solar system neighbourhood, the ones that are not planets, not dwarf planets and not moons? The organization has decided that most asteroids, comets and other small objects will be called 'small solar-system bodies.' Despite the establishment of these three distinct categories, there are bound to be grey areas. As technologies improve and more objects are found the International Astronomical Union will set up a process to decide which of the three categories are most appropriate for specific objects.