There is a prank that people do where you take your beer bottle and
hit the bottom of it on the top of someone else's. This causes the
beer in his/her bottle to foam up, overflow, and spill out the top.
What is the physics that causes this to happen and why doesn't your
own bottle spill over as well?

I have wondered about this myself.  In another 'beer bottle trick',
you fill a beer bottle almost all the way with water, and then hit it
sharply on the top with your palm.  The bottom of the bottle breaks
out, spilling the water on the floor, and leaving you holding a bottle
which is intact except that it's missing its bottom (just the flat
circular part).  (Long neck bottles work best because you can hold
them better in one hand while hitting with the other.)  Most online
references will tell you that the reason the bottom breaks out is
because you create a pressure wave which travels down through the
liquid till it reaches the bottom.  I didn't buy this argument, and
with the help of a friend who'd rented a high speed camera for other
purposes, I filmed the process at 6000 frames/second.  A bubble can be
seen forming in the bottom of the bottle after it has been hit.  This
is known as cavitation, and it occurs because the water can't keep up
with the sudden downward acceleration of the bottle, so a vacuum
forms.  Atmospheric pressure then slams the water back against the
bottom of the bottle, breaking it out.
I'm sure the same effect is operating in the overflow trick.  I don't
think a large cavity forms in that case, but the sudden downward
acceleration of the bottle causes a big drop in pressure.  Basically,
you're trying to 'stretch' the beer in the bottle, so the pressure
drops.  When the pressure decreases, the amount of CO2 which can be
dissolved in the beer decreases, so it comes out of solution.  This
causes the foaming which leads to overflow.  The reason the bottle
which does the hitting doesn't overflow is that during the impact,
that bottle experiences a sudden UPWARD acceleration, compressing the
beer inside so that the pressure increases, rather than decreasing.

I agree with racecar's analysis.  The principle might make a good
science fair project.  The upper bottle, moving downward with contents
in uniform motion, hits an obstacle and in the initial impact there's
a compression of the fluid contents at the bottom of the bottle. 
Under compression the solubility of CO2 actually increases, so there's
no sudden release of CO2 gas from solution at the bottle's lower
surface (although there will be some along the sides and top due to
the refraction of the compression wave as it passes quickly through
the fluid).

The lower bottle, however, upon being struck sharply moves downward
against the inertia of its contents.  This creates a decompression at
the bottom of the bottle's contents.  Solubility of CO2 drops
dramatically and a great deal of "fizz" is suddenly forced out of
solution.  The expansion of this gas pushes the fluid above up and out
of the bottle's mouth.

The "subtle" point here is why CO2 when in solution is more compact
than when out of solution.  The answer lies in an analysis of the
static electrical fields between molecules, but as a general
observation we are familiar with dissolving a considerable amount of
solids (such as sugar or salt) in water with little increase in the
volume of the water.  The same principle applies here.  A fairly
significant amount of CO2 can be dissolved in cool water with no
appreciable increase in the water's volume, certainly far less than
the volume of that CO2 by itself under atomospheric pressure.

regards, mathtalk-ga

I've seen this trick done where the bottle that is being hit is
standing on a table thus preventing downward movement of the struck
bottle. The beer fizzes out just the same but there must be different
theory at work to the one above.

I tend to favor an acoustic explanation...

Having had this trick played on me before, I also wondered about the
workings of this prank.

If you've ever tossed an empty beer bottle into a trash can, and it
doesn't break, you know that it makes a pretty good racket. My theory
is that the impact causes the bottle to vibrate at a certain frequency
which translates into the beer causing a ripple effect, which in turn
causes seperation of the dissolved CO2 at a faster rate than normal
ending with foam coming out of the top.

flajason-

You have an interesting theory, but how does it account for the
"striking" bottle, which must have the same frequency modes, not
fizzing over?

"I've seen this trick done where the bottle that is being hit is
standing on a table thus preventing downward movement of the struck
bottle. The beer fizzes out just the same but there must be different
theory at work to the one above."

I don't think a different theory is required.  Beer is quite
incompressible, so the bottle doesn't have to move very far to create
a dramatic drop in pressure.  If the bottle is resting on a
not-too-solid wooden or plastic table, it will still accelerate
downward upon being struck.  The fact that it moves only a mm or less
doesn't matter too much.  (It does matter some, as the motion is
limited, and in addition to accelerating the mass of the bottle, you
must also accelerate the mass of the table surface beneath it,
decreasing the magnitude of the pressure drop--and if you try it,
you'll see that a hand-held bottle will fizz over more than one
sitting on a table when both are hit with the same force).  If the
bottle is resting on something really solid, like a concrete floor,
the prank does not work.  A little fizzing is observed in each bottle,
but the effect is no bigger in the struck bottle than in the striking
one.

I wass motivated by ffxdir-ga's comment to try an experiment.  I
placed a bottle on  a hearth, rather than a table.  Of course the
hearth is typically less flexible than a table top.  While a few
bubbles appeared from striking the top of the bottle, there was no
overflow.

regards, mathtalk-ga

This may have a full-size counterpart in that Lakes Monoun and Lyos in
Africa several years back had sudden CO2 release "events" that killed
around 1700 people.  Both lakes are connected with underground
volcanic activity that saturate the waters with CO2.  The release
occurred during a drop in air pressure coupled with a seismic event. 
What I'm guessing here is that the drop in pressure above the water
meant that the saturated CO2 in the lake was now out of balance with
the the air pressure (Henry's Law); coupled with the vertical rise in
the water (that was proposed in an earlier posting) by the seismic
activity creating a lower pressure at the base of the lake caused the
CO2 to be released from the water in one big, rapid release- just like
the beer bottle since beer is over-saturated with CO2 when it's
bottled (sodas- and I think beers- are pressurized around 2 atm before
capped). If I remember, this trick won't work on warmer flat beer, it
needs to be a fresh cold one. Based on the lake example, I'd have to
put(most of) my money on the sudden decrease in pressure due to upward
movement (however slight) of the beer- but then again that's just a
reasonable guess... they're still unclear as to exactly why the lakes
burped, too.