Whether it's a bathtub draining or a spinning tornado, vortexes are caused by downdrafts that form when high pressure and low pressure interact. The conflict results in the creation of a swirling suction that, in the case of a tornado, can be very destructive.
A bathtub drain's vortex is formed by gravity pulling the water down into the pipe at the same time the air in the pipe is trying to push it up. The two opposing forces cause the water to start swirling. As the speed increases, the suction is created and the water swirls down the drain. Any material caught in the vortex will spin around the drain, gradually spiraling into the pipe.
A tornado's vortex is created by a super-cell thunderstorm. The storm creates a downdraft which starts to rotate, creating the vortex. A cloud wall is formed when the vortex starts pulling the cloud mass into the vortex, creating the familiar funnel shape. As more warm air from the thunderstorm is pulled down into the funnel, the vortex strengthens. If the funnel touches the ground, it becomes a tornado.
Eventually the air in the downdraft cools and the funnel rotation slows. The vortex starts to break down and eventually dissipates. If the thunderstorms are severe enough and there is enough moist warm air in the atmosphere, multiple vortexes may be created.