The hang-glider flies just like any aircraft. Air moving over the wings has further to go over the upper curved surface and so travels faster. This causes the air to drop in pressure. With a lower pressure above the wing than below it, the wing is forced upward. This is called Lift. Think of it like a large paper plane.
Control of the Glider is initiated by the pilot shifting his weight. Forward initiates a dive and increased speed, backward a climb and reduced speed, and left and right initiate turns.
How the glider turns these inputs into movement is somewhat technical. A glider wing design is very complex. if you would like to know more search these terms: billow shift, sweep back, reflex, washout, center of pressure.
A glider can launch into the air by three main methods; Hill, Tow and Aerotow. During hill launching the pilot will run off a hill, ridge or cliff to get airborne; while towing and aerotowing both pull the pilot and glider via a cable. The first being attached to a winch on the ground and the latter attached to a powered aircraft such as a microloight.
Other than Powered flight a glider can stay airborne and even navigate long distances through utilising two main sources of lift: Dynamic and thermal. Dynamic lift is present when fast moving air is forced up over high ground such as a hill or ridge. This creates a lift band of rising air up wind of the feature that pilots can fly along. Thermal lift is generated as the sun warms the ground. Air close to the ground forms a bubble much like a lava lamp which breaks away and rises. glider pilots find these bubbles and circle in them climbing as high as the base of the clouds.
A UK competitive sport exists where pilots compete in achieving the furthest distance away from the launch point un-powerd. These events can cover hundreds of kilometers.