Sharia forbids earning or paying interest on loans. So Sharia finance is about the ways to get around this prohibition. Sharia finance also concerns prohibitions on using money to finance the pork or alcohol industries.
The first is not a problem for dividend or capital gains earnings because they clearly involve risk which is a requirement of Sharia investments. The problem arises with home mortgages and savings accounts.
Home mortgages are handled by having a Sharia bank buy the relevant house and then sell it back, on a monthly payment schedule, to the mortgagee plus a rental fee on the use of the house. The net result is identical to a mortgage loan.
A savings account in a Sharia bank is one where the deposited money is invested in only Sharia investments. The returns are dividends and capital gains which are put in a fund that pays out the regular market savings account interest rate. Again, this is not really very different than a standard bank savings account except for the way the funds are invested (definitely not as secure as a bank savings account and certainly not insured by any government.)
All other Sharia finance is based on equity funds that exclude interest bearing instruments, like bonds, and avoid stocks in inappropriate (pork and alcohol) corporations. (Think socially responsible funds with an Islamic agenda.)
The multiple tools of hedge funds... options, puts, calls and arbitrage contracts are nearly all well suited to Sharia finance (excluding bets on interest rate markets such as Libors.)
The Sharia prohibition on interest was appropriate at the time of Mohammed, as Christianity took the same position. (Jews in both Moslem and Christian societies were relegated to banking.)We now see interest as a rental on money composed of a mix of risk, market pricing (national price index) and the supply of funds.