What is Islamic Banking? What is "Islamic banking," and how is it similar to and different from banking as it occurs in the U.S.?

I have heard Shariah Law prohibits usury, the taking of interest on loans. How do Muslim bankers justify the prohibition of interest? Are there any prominent Muslim economists, contemporary or older, who justify their prohibition of interest not only from a religious perspective?
Both another answer and the OP have linked to the wikipedia article on the matter. I think the following passage from the article sums adequately the situation (bold mine):


  "The term “Islamic banking” refers to a system of banking or banking activity that is consistent with Islamic law (Shariah) principles and
  guided by Islamic economics. The contemporary movement of Islamic
  finance is based on the belief that "all forms of interest are riba
  and hence prohibited".[48] In addition, Islamic law prohibits
  investing in businesses that are considered unlawful, or haraam
  (such as businesses that sell alcohol or pork, or businesses that
  produce media such as gossip columns or pornography, which are
  contrary to Islamic values). Furthermore, the Shariah prohibits
  what is called "Maysir" and "Gharar". Maysir is involved in
  contracts where the ownership of a good depends on the occurrence of a predetermined, uncertain event in the future whereas Gharar
  describes speculative transactions. Both concepts involve
  excessive risk and are supposed to foster uncertainty and fraudulent
  behaviour. Therefore, the use of all conventional derivative
  instruments is impossible in Islamic banking. In the late 20th
  century, a number of Islamic banks were created to cater to this
  particular banking market."
  
  "Islamic banking has the same purpose as conventional banking: to make money for the banking institute by lending out capital while
  adhering to Islamic law. Because Islam forbids simply lending out
  money at interest, Islamic rules on transactions (known as Fiqh
  al-Muamalat) have been created to prevent it. The basic principle of
  Islamic banking is based on risk-sharing which is a component of trade
  rather than risk-transfer which is seen in conventional banking.
  Islamic banking introduces concepts such as profit sharing (Mudharabah), safekeeping (Wadiah), joint venture
  (Musharakah), cost plus (Murabahah), and leasing (Ijar)."


I can also offer my professional experience on the matter, which says that some times these are used as merely veils for what is essentially a conventional loan earning interest.

As a side remark, Christian institutions like the Catholic Church never officially lifted their condemnation of "usury"/charging interest -they just don't actively enforce the doctrine. (As far as I could find, the most recent papal discussion of usury occurred in Pope Benedict XIV’s encyclical of 1745, Vix pervenit.)