The base level convection current comes from heating a substance, such as water, on a burner. As the water gets warmer it becomes less dense causing it to rise and the cooler water to sink. As the heated water rises, it slowly warms the water around it as well. This current eventually heats all of the water if the heat source stays consistent. If the heat source is turned off, the cooler water will win out, and the current will cool the warm water.
On a larger scale, convection currents affect the air around you. Cool air with pollutants becomes the heaviest air. While the warm air rises past the cool air, it still maintains some of the pollutants. Instead of standard convection allowing the heat to rise to the top of the atmosphere, it stays weighted down by the additives. This keeps the best, cool, clean air, above the heated air. This causes a temperature inversion, stopping the standard convection process in the air.
This convection actually happens within the magma of the Earth, technically a liquid. The super heating and cooling of the magma in the Earth's mantle, through convection, causes the magma to move below the surface of Earth. As it presses on the Earth's crust it can find cracks, pushing up the solid land forming mountains. This also pushes on the tectonic plates causing seismic activity and earthquakes. This convection current, or heat transfer, is the first step in moving mass solids.
No matter what type of convection you are looking at, it is the movement of heat. On a global level, cellular level or subatomic level, the change in heat causes motion controlling everything from air temperature, water temperature, and even the movement of water in the oceans. Convection also controls the basis of our heating and cooling systems in buildings.