There are a few fundamental differences between oars and paddles.
One main difference is the position of the user relative to the direction in which the vessel is moving.
Oars are used to propel the boat in the opposite direction from the direction the rower is seated. Therefore, the person who rows the boat travels backward.
Paddles propel the boat in the same direction the paddler is facing. Therefore, the paddler travels forward. (Source)
Oars are used primarily for rowboats and sculls, although there are other popular oar-driven vessels.
Paddles are used primarily for canoes and kayaks. Paddle design even differs between canoes and kayaks. Canoe paddles have long handles with one blade on the end. Kayak paddles have blades on both ends.
Oars are attached to the boat using oarlocks, which come in many shapes and sizes, and can be adapted for a number of different types of boats. Because they're attached, the motion of oars is controlled by the legs, knees and arms.
Paddles are not attached to the boat, so they're held by the hand, or both hands, depending on the situation. The strokes of a paddle are controlled by the user's torso.
Both oars and paddles are made in a number of different materials. Varieties include wood, fiberglass, plastic, as well as nylon and composites. They also both come in different sizes, shapes and designs. They're tailored to the size of the user, the number of users per craft, and the shape or purpose of the boat. Paddles and Oars offers information, descriptions, pictures and recommendations for choosing which works best in any given situation.
My thanks go to @Erik for providing additional pertinent information regarding oars:
Oars give a much more powerful stroke than paddles due to their increased leverage. This is why bigger heavier crafts typically use oars instead of paddles.