Why are wing tips curved up on commercial airliners?

The curved part of the wing tip is called a winglet a term used by Richard Whitcomb, a well known aerodynamicists that worked for NASA nasa.gov/richard_whitcomb, who refined the concept. These winglets are in place to help reduce the drag. More specifically the induced drag caused by the wing tip vortices that are created by the pressure difference between the top and bottom surfaces of the wing.

Airfoils create lift by creating a pressure difference on the wing from the difference in velocities on each surface. According to Bernoulli's principle a high velocity will create a low pressure and a low velocity has a high pressure, the airfoil creates this velocity difference because of its shape making air over its top faster than its bottom, thus creating lift. Anyone familiar with the concept of equilibrium will understand that air at high pressure will want to move to an area of lower pressure, this movement is what happens at the wing tips of aircraft as seen in the above picture. This movement creates what is known as a vortex which increases the pressure on the top of the wing which is undesirable to create lift. 

The way these winglets cut down on the drag is to disrupt or block some of the movement of air from the high pressure side to the low pressure side and thus creating a smaller vortex and less induced drag. 

There are other ways to reduce induced drag such as an elliptic lift distribution which is derived from the Kutta-Joukowski theorem based on lift from circulation, a concept you'll need to read the book to fully understand. After all the derivations it was found that the elliptic lift distribution provides the least amount of induced drag. To create an elliptic lift distribution you can create an elliptic planform as seen below. This, however, is hard to manufacture and thus expensive. An alternate option that is close is to taper the wing, so the point in which the wing attaches to the fuselage is the root chord and the end of the wing is the tip chord length and in case you were wondering the ideal ratio is tip/root of about 0.3. Most commercial airliners use a tapered wing with winglets. Winglets have been around since the 1950's but due to the increased cost of fuel they became popular again in today's industry.