Why is 1 AU the distance between the Sun and the Earth? Why 1 AU is defined as the distance between the Sun and the Earth? (approximately if you like to be precise) 

An astronomical unit (abbreviated as AU, au, a.u., or ua) is a unit of length equal to about 149,597,870.7 kilometres (92,955,807.3 mi) or approximately the mean Earth–Sun distance.

Shouldn't astronomical units be defined within metric units (that is,  $10^x$), so we can understand massive distances a little easier?
 A: 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.

A: 1 AU is the mean distance of the Earth from the Sun, by definition.  Its value is approximately 1.5 10^11 meters.
A: In the 1800's the AU was connected to: the time average of 1 divided by the Earth-Sun,
and that is the reason Gauss's constant occurred (until 2012) in the 
calculation of the AU. The reciprocal of the distance was used because it
has less of a linear tread and also because it is not as dependent on
the eccentricity (which has a large ~linear trend over the last centuries).
For many decades the specific value for the AU was a representation of
the strength of the Sun's gravity field converted to a length in meters,
thus it was a measure of G but connected to the large masses of the
planets and the distances measured by radar to them, so its precision
can not be used with objects with small masses since there is no
way to measure the planets masses in kilograms accurately.
Then in about 2012 the astronomical union fixed the AU at exactly
149597870700 meters. Thus its value does not currently reflect the distance
from the Earth to the Sun in any way other than being close to historical
guesses of it.
 It is telling that a "Google search" reports to the public
the near opposite of what I know to be true as stated above.
Thus for science much of the information on the internet is
more harmful than helpful.
A: The definition of the AU does not involve the earth at all.
Now for the question, astronomy makes a heavy use of trigonometry, where angles are computed from distances. So a lot of computations involve the Earth-Sun distance. Normalizing the semi-major axis to 1 is a time saver for such calculations.
I agree with your point about the metric system. I guess that it may become commonplace when space travel becomes afordable. How many Gm/L (gigameters per liter) goes this spaceship ?
