At the moment, they're just an extra commodity to trade with. There are no dividends/profit shares involved, and we don't have a stock exchange where you can buy or sell them.
A faction owner can issue shares, then sell them to earn extra credits. The combine value of the shares are the faction value split equally between the number of shares, however, the market value of the shares is dictated by the players (see the Centrepoint Database
for current averages).
Whoever owns 51% of the shares of the faction has ownership control. Anyone with 49% or less has the ability to view the faction's status. These are basic statistics that show the faction's value, their percentage increase, any threats of dissolution due to inactivity/bankrupcy/disinterest and lists every other shareholder of the faction.
That's a very simplified description, if you have any more questions, let me know :)