I am building a computer. I plan to purchase a separate video card
such as an nVIDIA GeForce 6600GT. I will also be purchasing a
motherboard. However I've noticed that motherboards can vary
significantly in price depending on the integrated graphics chipset
included. (e.g. nForce, GeForce 6100, ATI XPRESS 200)

If I'm buying a graphics card to attach to the motherboard, does the
quality of the graphics processing on the motherboard matter? Would I
be wiser to choose a higher performance onboard graphics chipset or
just spend the money on a better video card?

No.

There is no such thing as a high performance onboard graphics chipset.
They are all basic. What i mean by basic is video rendering,
resolution output, dvd decoding, that kind of thing.

When building a PC if you have a PCI-e or AGP card in your system,
onboard video would actually be disabled (Or you can disable it
through the BIOS)

If you buy a 6600GT, The onboard video will do nothing for
performance. Its seperate.

Also remember, chipsets arent just graphics. They do have other
features such as 7.1 Digital Audio, SATA Raid, NV Firewall, etc.

It actully depends on your requirement. I would say that onboad
display card will do as it is almost equal to separate card. Only make
sure how much memory your vendor is giving on the onboard card.
Typically you would get 64mb on the on board display card which is
great. If you are a gaming frek and love to play games then you may
consider separate card as some games work on 128mb display memory
howerver most work on 64mb. If I were you would go for onboard
integrated card which would save money and less hassel

Umm yeah, first off whether a game will run or not is not based on the
video cards memory but on your systems overall capabilities.  Why does
it not?  It doesn't because every game I've played (which is most
games) has settings to lower the texture settings for people with
video cards that don't have huge memory buffers.  Unless your a game
guru this shouldnt matter to you.  The biggest factor in running
today's games is system memory (RAM).  Most of todays games require at
least 256MB but recommend 512MB and this is quickly doubling.  Keep in
mind this is system memory not video card memory.  Although the more
the better in either case.  Of course there's a point where too little
is not enough for video cards, so if you're on a budget concentrate on
the CPU, RAM, and video card for a gaming rig.
  Games these days are quite demanding on your CPU as well as the
video card, but if your system is just plain old or too slow it wont
run fast enough to be enjoyable.  If you plan on playing alot of
todays "demanding" games then you will need to opt for an add-on video
card.  Many games these days require tons of processing power which in
turn requires a powerful video card.  On-board cards are fine for
video editing, multimedia, and basic use, and they can play most games
these days, just not nearly as fast as a video card can.  If you do
play alot of games and are buying a video card, pay no attention to
the on-board video processor.  Not sure what exactly you are looking
for but the "N-Force" chipsets are outstanding as far as overall
system performance, reliability, and lots of features goes.  You can
see the list of the available Nforce models here:
http://www.nvidia.com/page/nforce4_family.html. Next time you want
advice on a purchase make sure you are as specific on your needs and
wants as possible because without knowing what you want makes it
impossible to give helpful advice.

No.  As stated you would be disabling the on board chipset to use your 6600GT.

Another thought:  If you want further graphics expansion in the future
and are buying a PCIe graphics card, look at a motherboard that
suports either SLI or Crossfire technology.
These basically allow you to buy your PCIe video card (the 6600GT
supports SLI) and add another (identical) card in parallel in the
future.  Generally you will see upwards of 60% increase in performance
over a single card.