Order routing is the process by which an order goes from the end user to an exchange. An order may go directly to the exchange from the customer, or it may go first to a broker who then routes the order to the exchange.
"Smart" order routing attempts to achieve best execution of trades while minimizing market impact. It is designed to help firms in an increasingly fragmented market to search for hidden liquidity, find opportunities in dark pools and use algorithms to maximize results without moving the market. 
In stock trading, it is a common practice among brokerage firms to route orders to certain market makers, who then "rebate" 1 to 4 cents per share back to the brokerage firm in exchange for the directed order flow. This practice is known as "payment for order flow" (PFOF).