I am a college student in an introduction to business class. We are
studying the exchange rates between different currencies. I have
chosen the Dollar to the Euro as my choice of exchanges to study. I
see that on Sept. 7, 2006, the Exchange rate was 0.7798. The "bid" was
0.7798. The "ask" was 0.7799. My first question: Who does the
"bidding," the "asking"? My second question is related: How do the
bidders and askers decide what they are willing to accept? In other
words, what are the major factors that influence the price of the
dollar against the euro?

As far as I remember, the prospective seller is asking 0.7799, but the
prospective buyer is only offering 0.7798, so you don't have a sale.
It does, however, give an indication as to the value. Your second
question is the tough one.

Foreign exchange dealing is very similar to what you experience at the
exchange dealer's booth in an airport.  The dealer has two prices: he
will buy a currency at his "bid" price and sell it at his "ask" price.
 Obviously, he will buy at a lower price than that which he is willing
to sell for.
In the forex market between financial institutions, the dealers
operate the same way, but at much finer margins, as your example
shows, but they are dealing in tens or hundreds of millions. 
Somewhere in the market, there is an institute that will sell (its ask
price) at the bid price of another institute.  Of course, the bid and
ask prices are only indicative, the forex dealers can buy or sell at
whatever price they choose.  For the most part, dealing is
speculative:  trying to make a profit on day trading; making educated
(?) guesses about the way the market will move by anticipating
economic developments, studying charts of the movement of the exchange
rate.
When a bank makes a forex contract with a customer, the margin will be
greater, the dealer's bid price lower, so he can sell the currency at
a lower price than he would if he had bought it at his bid price for
the interbank market.

What influences the Euro-dollar market?  Economic indicators, changes
in interest rates, political developments.
If the US announced that it was getting out of Iraq, the dollar would
probably rise, because a lot of people think this would be a good
thing (emotions), but more significantly because the US would not have
to keep increasing its debt, which should be good for the economy.  If
the Fed announces that it will increase interest rates, the dollar may
rise, because it is more attractive then to hold dollars.

But don't quote me on this latter bit.  I hope the first part helps
you understand bid and ask a little better.

just to add - these rates you see on wallstreet journal do not
represent the rates you get when you want to change money. they are
rates for huge transactions of some millions. see www.oanda.com for
some things and past forex rates. At http:// you get tons of historic
exchange rates. http://fx.sauder.ubc.ca/