Author: many

This question typically arises during debates regarding whether a
government should spend money on astronomy.  There are both pratical
and philosophical reasons that the study of astronomy is important.

On the practical side...

Astronomical theories and observations test our fundamental theories,
on which our technology is based.  Astronomy makes it possible for us
to study phenomena at scales of size, mass, distance, density,
temperature, etc., and especially on TIME scales that are not possible
to reproduce in the laboratory.  Sometimes the most stringent tests of
those theories can only come from astronomical phenomena.  It must be
understood that these theories influence us even if they don't tell us
that we can invent new things, because they can tell us that we can't
do certain things.  Effort spent on astronomy can prevent effort
wasted trying to come up with antigravity, for instance.  

Astronomy provided the fundamental standard of time until it was
superseded by atomic clocks in 1967.  Even today, astronomical
techniques are needed to determine the orientation of the Earth in
space, e.g., <URL:http://www.usno.navy.mil/>.  This has military
applications but is also needed by anyone who uses the Global
Positioning System (GPS).  Furthermore, it may be that millisecond
pulsars can provide an even more stable clock over longer time scales
than can atomic clocks.

Closely related is navigation.  Until relatively recently (post-WW II)
celestial navigation was the ONLY way in which ships and aircraft
could determine their position at sea.  Indeed, the existence of
navigation satellite systems today depends heavily on the lessons
learned from aspects of astronomy such as celestial mechanics and
geodesy.  Even today, in the UK, RAF crews and RN officers need to
learn the rudiments of celestial navigation for emergency purposes;
until the late 1990s so did US Naval officers.

Astronomical phenomena have been important in Earth's history.
Asteroid impacts have had major effects on the history of life, in
particular contributing to the extinction of the dinosaurs and setting
the stage for mammals.  The Tunguska impact in 1908 would have had a
far greater effect if it had occurred over London or Paris as opposed
to Siberia.

The debate over the magnitude, effect, and cost of greenhouse warming
is motivated, in part, by research on Venus.  Astronomy has prompted
study of the Earth's climate in other ways as well.  The study of the
atmospheres of other planets has helped to test and refine models of
the Earth's atmosphere.  The Sun was fainter in the past, an important
constraint on the history of the climate and life.  Understanding how
the Earth's climate responded to a fainter Sun is important for
evolution and for the progress of climate modelling.  More generally,
there is weak evidence that solar activity influences climate changes
(e.g., variations in sunspot cycle, the Maunder minimum, and the
Little Ice Age) and therefore is important in the greenhouse warming
debate.  (This is by no means proven by current evidence but *may*
prove to be important.)

The element helium was discovered (in a real sense) and named, not by
chemists, but by astronomers.  In addition to making many birthday
parties more festive, liquid helium is useful for many low-temperature
applications.

Solar activity affects power-grids and communications (and
space travel).  Prediction is therefore important, indeed is
funded by the U.S. Air Force.

Many advances in medical imaging are due to astronomy.  Even the
simple technique that astronomers used for decades, of baking or
otherwise sensitizing photographic materials, was slow to catch on in
medical circles until astronomers pointed out that it could reduce the
required x-ray dose by more than a factor of 2.  Many of those now
involved in some of the most advanced developments of medical imaging
and imaging in forensics were trained as astronomers where they
learned the basic techniques and saw ways to apply them.  More
recently, image reconstruction of the flawwed Hubble images led to
earlier detection of tumors in mammograms (see back issues of Physics
Today).

While we don't yet have a good method for predicting earthquakes, the
techniques of Very Long Baseline Interferometry are used routinely to
measure ground motion.  

Interferometry has also led to the development of Synthetic Aperture
Radar.  Today SAR is used for earth remote sensing.  Applications
include mapping sea ice (safety of ships, weather forecasting) and
ocean waves (ditto), resource location, agricultural development and
status checks.

Jules Verne would never have written "From the Earth to the Moon"
without astronomy.  Astronomy helped spawn science fiction, now an
important component of many publishing houses and film studio
productions.

There has been a complex interplay between scientific, military, and
civil users, but astronomy has played an important role in the
development of such things as security X-ray systems (like those at
airports), electro-optics sensors (security cameras, consumer video
cameras, CCDs, etc.), and military surveillance technology (like spy
satellites).


On the philosophical side...

Perhaps the most important aspect of being human is our ability to 
acquire knowledge about the Universe.  Astronomy provides the best
measure of our place in the Universe.

In this century, the ability of astronomy to test General Relativity
led directly to Karl Popper's distinction between science and
pseudo-science and from there to the way intellectuals (at least) look
at science.  Astronomy's support of modern physics (such as quantum
mechanics) in this century had have important influences on general
philosphical and intellectual trends.  The "Earthrise" photo, of the
Earth rising over the Moon's horizon, from an Apollo mission is often
credited as being partially responsible for driving environmental and
"save the planet" impulses.

In previous centuries, astronomy led to Copernicanism and subsequent
"Principle of Mediocrity" developments---that the Earth, and by
extension, humans, is not at the center of the Universe.  Eliminating
geo- and human-centred perspectives was a major philosophical leap.
Astronomy's support of a mechanistic universe in the 19th century had
important influences on general philosphical and intellectual trends.

In general, but certainly more vaguely, the last century of astronomy
has provided many supports to the view that the scientific method is
capable of answering many questions and that naturalistic thinking can
explain the world.  Thus, scientists can answer many creation
questions (e.g., where metals come from, why the Sun shines, why there
are planets).