I am looking to get a better understanding of what the US trade
deficit really means.  More importantly, I'm wanting to know how
exactly we end up owing other countries money during the process.  I
realize that running a trade deficit by definition means that a
country's imports exceed their exports, but aren't those imports
bought and paid for?  If not, how does the process by which the
financing and actual purchase of these goods take place such that we
end up owing money to other countries?  Lastly, which countries own
the major share of U.S. debt as of this year?  Thanks.

In effect the US trade deficit is financed by loans from other countries.

Typically they buy US Treasury Bonds and hold them, although any
financial instrument will do.

The total Balance of Payments falls into parts, 
  the Current Account - goods and services
  the Capital Account - borrowing and long term inward investment

Overseas investors can buy USA property and companies, that is long
term inward investment, and they can buy 'financial instruments' or
simply deposit the money in the USA - the last is often called 'Hot
Money'.

Currently I believe that the major holders of US debt are Japan and
China, although many countries prefer to hold USD 'assets'.

http://usinfo.state.gov/eap/Archive/2006/Feb/17-883919.html

||Washington -- Federal Reserve Chairman Ben Bernanke played down
fears expressed by U.S. lawmakers that China might shake the U.S.
economy by selling significant chunks of the U.S. debt it holds.

In his first appearance before the Senate as the head of the U.S.
central bank, Bernanke was pressed by Banking Committee members on the
question of the U.S. economy's vulnerability to changes in China's
U.S.-dollar denominated assets.

But Bernanke said he is not "deeply" concerned about the issue.

"I don't think that the Chinese ownership of U.S. assets is so large
as to put our country at risk economically," he said February 16.

At the end of 2005, China, with $820 billion in such assets, was the
second-largest holder of U.S. debt after Japan, which held about $10
billion more.

A day earlier, answering similar question from a House of
Representatives committee member, Bernanke said China is holding only
a small percentage of the overall U.S. debt, which in addition to U.S.
government bonds includes government-sponsored enterprise and
corporate debt securities. ||

Basically one finances a Trade Deficit by :-
  a) Selling the family silver to overseas buyers
  b) Borrowing from overseas lenders

In my view, the situation is not that fragile, as borrowers like to
lend in $US, even if it is not to the USA - note the Eurobond markets.

Having continuous trade deficits is not a sensible idea, as one will
eventually hit a point where you 'have sold the farm' or interest
payments are crippling
- or one is tempted to use inflation to shrink the real value of the debt

To reinforce the point made by frde: The current account and the
capital account are always in balance - if you have a surplus frome
trade, goods flow out of the country, you have a surplus on the cash
account - money flows in.

You ask "I
realize that running a trade deficit by definition means that a
country's imports exceed their exports, but aren't those imports
bought and paid for?" Yes, these foreign goods are paid for with
US-Dollars. If I swap my Euro against Dollar, I loan the armerican
economy money to buy foreign goods with.In other words: other
countries not only give the US a considerable part of their output,
but also the money to pay for it. ;-)