Nature will always try to achieve equilibrium, or the state of all things being equal. A convection current is a movement of objects and forces--down to the molecular level--to achieve that equilibrium. Visualize this experiment: you have two jars connected with a tube at the bottom, with a shutoff valve in the tube. Fill one jar with water almost to the top, leave the other jar empty, and then open the valve. Water will flow through the tube from the filled jar to the empty jar until the water level is the same in both jars. The flow in the tube represents a convection current. This powerful force is a common denominator of all convection currents.
Temperature differences inside a body of water or air causes molecules to move. A current occurs because hotter molecules will try to travel to the top, and colder molecules will sink to the bottom. Duke University demonstrates air currents by having a room with a cold ceiling and a hot floor (see reference 2). Another powerful example of a convection current in air is a tornado. As air in the upper atmosphere cools, it trys to sink to the ground. In the process of falling, it forms a vortex. This vortex is seen by the rain drops that are carried along by the tornado. An example of a convection current in liquids is a lava lamp. As the wax at the bottom is heated, it will rise to the top. Once at the top, the wax cools and falls back down.
If you look at a photograph of a spiral galaxy, notice that it is much like the cross section--looking at it from the top--of a vortex or tornado. The astrophysical forces of gravity are not yet fully understood, but research is ongoing. Part of that research is investigating what roles gravitational convection currents play in forming a galaxy.
Convection currents are responsible for earthquakes. According to the University of Washington, earthquakes occur because forces build up inside the earth. The comparison of the core movement to the mantle is similar to the convection currents inside a lava lamp. When the forces overcome the force of the mantle holding together, the surface moves to neutralize the forces. Again, this is equilibrium in action. Nature is trying to take too much internal pressure, and move it to weaker areas of the crust.