A helium-filled balloon for a birthday party is being brought home in
a car. As the car accelerates away from a stoplight, in which
direction does the balloon move relative to the car? Explain.

The baloon, if it was not touching any part of the car, would
naturally attempt to stay in the same (horizontal) place while rising.
So, if the car moved forward, the balloon would appear to be moving to
the back of the car.

Q:The problem is I have a Helium balloon tied to the back seat of my
car. The Balloon is not touching anything but the string that it is
tied to. When I slam on the gas pedal of the car and start
accelerating the balloon drifts to the front of the car. The faster I
accelerate the more the balloon leans to the front of the car, until I
reach cruising speed."

A:When you accelerate, there is a ?pseudo-force? that pushes most
things to the back of the car ? this is the feeling of being pushed
back in your seat, when you accelerate and also throws you against the
seatbelts when you brake suddenly. This is not a real force ? it is an
effect of your inertia (your tendency to remain in the same state of
motion) hence the name pseudo-force. Pseudo-forces are present anytime
your frame of reference accelerates; "centrifugal forces", due to the
constant acceleration of circular motion are another famous, familiar
example. They even affects the air in your car ? so when you
accelerate, the air tends to move backwards a little ? not so much
that people on the front seat suffocate, but enough that the air gets
a little more dense (and higher pressure) the further back you go.

Source: http://physics.about.com/cs/acceleration/f/heliumballoon.htm
As a helium balloon floats due to Archimedes' Principle it rises in
the direction reduces the pressure of the air fastest. In normal air,
the pressure drops as you go directly upwards (only slightly, but the
balloon feels this slight force) but in your car, when the pressure
builds up briefly at the back, the balloon ?rises? forwards. The
balloon will stand vertically if you have constant speed and to go
backwards when you brake except the pseudo force on it when you brake
sharply may overwhelm the pressure gradient and make it go forwards
anyway.

When I explained this to my test helium balloon, it took it all in and
then dived under the gifts in back seat where it remained, quivering,
until we got home. It has been refusing to come out since and keeps
shrieking in that high-pitched helium voice something about how
centrifugal force is an illusion and there is only centripedal force.
I tried to tell it that this doesn't apply here, but to no avail.

The helium filled balloon is lighter than air so I would expect that
air resistance and air movement would have a large effect.

As Cynthia noted, when the car accellerates the balloon will tend to
stay in the same (horizontal) place.  So the balloon would be moving
backwards relative to the car.  But this ignores the affect of air
resistance and air movement.

As Canadianhelper noted, the accelleration would also compress the air
toward the back of the car.  But when that accelleration ended (when
the car reached cruising speed) the air would drift back forward to
equilize the air pressure in the car again and the balloon would drift
with it.

But all of the above could be overwhelmed by the circulation of air in
the car.  Certainly if the air conditioning or heat is on in the car
or one window is open even a crack, this could create air currents in
the car that would cause the balloon to drift around.  And these air
currents can sometimes persist a lot longer than intuition would
generally expect.  I have seen old helium balloons that have achieved
neutral buoyancy in a closed room, neither rising to the ceiling or
sinking to the floor but just floating around at a constant elevation
between floor and ceiling.  These balloons can slowly drift around the
room, apparently on air currents to gentle to feel.

I?ll buy Canadianhelper?s explanation.  I have the impression that
something similar happens with warm and cold air in my car, that the
heavier cold air moves to the outside inside side of the car in a
curve.  (I tried to avoid that phrasing, but then rather liked it. :) 
That is, when I take a left turn, I feel cooler air.

The balloon has very little inerta so accelerating it is easy. I doubt
that it would stay in the same place for long. Even the air in the car
might be dense enough to accelerate the ballon.

The reason the balloon appears to come forward is that under hard
acceleration most cars. Especially older ones or non sports models
will pitch back. The rear of the car "squats" Most people are so used
to the sensation they don't notice it.

My guess is that the balloon string is staying almost perfectly
verticle while the automobile is actuialy leaving the horizontal under
acceleration.

Okay, so I went and got a helium balloon and drove around. The balloon
indeed moved forward when I accelerated and backward when I braked.
Canadianhelper's stated reason is the correct answer.

Now I have a spare SpongeBob balloon in my car...