I want to know what stakes are, in the context of a publicly traded
company. I would like to know how you get to own stakes in a company,
and how once you own them, how they produce money for you.  For
example, on Forbes.com, Prince Alwaleed Bin Talal Alsaud has stakes in
a company called Citigroup.  So how does he make money form these
stakes then?  Does he just passively own part of the company or does
he have to play an active role in running / managing it?

Hi,
Having a "stake" in something in this context is a participation,
owning shares in a publicly traded company.  You and I can have a
"stake" in Citigroup by buying the shares, though I suspect that the
prince has a really significant number of shares.
According to the dictionary, this usage of "stake" is a gambling
expression:  "that which is wagered in a game, race, or contest.

The prince may just passively own a part of Citigroup, earning
whatever dividends are paid.  If he has really significant number of
shares, he may be able to claim a seat of the board of directors,
which in principle represents the shareholders, but does not manage
the company.

Citigroup is one of the largest financial groups in the world.  Here
is its website:    http://www.citigroup.com/citigroup/homepage/

Prince Alwaleed Bin Talal Alsaud as of 7/02, owned 266,141,000 shares
(5.116%) of Citigroup (source: Bloomberg).  While he has considerable
clout in terms of voting via proxy, he as an investor does not play an
active role in the company.  As the second largest single investor, he
clearly will have access to Citigroup's top management and may offer
his advice, but the management team is by no means obligated to follow
his commands.

The Prince does not hold a board seat and thus cannot wield any
executive control.  His investment is a passive one that yields
dividends and more importantly equity appreciation.

With dividends currently at $1.76 per share per year, the Prince can
expect annual yield in dividends of $468,408,160.