Why is 1 AU the distance between the Sun and the Earth?
Using the distance between the Sun and the Earth, at least for distances within the Solar system, just gives a better feel for the scales involved.
You can't really imagine a distance of, say, 1000000000 kilometers -- or at least I can't. (I deliberately didn't include commas in that number, to illustrate the point.)
But using a concrete physical distance creates a kind of mental anchor, and makes the relative scale easier to visualize.
Tell me that Neptune is about 4.5 billion kilometers from the Sun, and I think "Wow, that's a really big number". Tell me that it's about 30 AUs from the Sun, and that's something I can fit into a mental image. One AU is still unimaginably long, but the ratio of 30 AUs to 1 AU is easy.
On the other hand, if you want to do physical calculations (say, calculating the orbit of some body under the influence of various gravitational fields), then it makes more sense to use metric units (meters, kilometers). The universal gravitational constant G is expressed in units of m3·kg-1·s-2; it could be expressed with an AU as the length unit, but I've never heard if it being done that way.
Basically, AU is used to express distances for a human audience; meters and kilometers are used for calculations.
Update : ghoppe comments:
Actually, ephemerides have been often calculated in astronomical units and not in SI units because neither G nor the mass of the sun can be measured to high accuracy in SI units, but the value of their product is known very precisely due to Kepler's Third Law. The value of AU depends on the product.
1 AU is the mean distance of the Earth from the Sun, by definition. Its value is approximately 1.5 10^11 meters.