The bands are due to a combination of heat transport from the equator to the poles and the planet's rotation. If a planet with an atmosphere does not rotate, then this heat transfer will occur by having winds establish themselves that blow from the equator to both poles at high altitudes, they dump their heat at the poles, and return to the equator by blowing back that way at low altitudes.
Now, think about a small, slowly rotating planet like Earth. You end up with about four different interlocking subcells (Hadley cells) due to an interaction with the coriolis force. These subcells have winds that blow towards the poles and towards the equator at different altitudes but, due to the coriolis force, they also have east and west components, too. These subcells mesh just like gears, and that's why we have some lattitudes with easterly winds and some with westerlies at the surface. The trade winds are a good example. Sailors knew to go N or S to find winds that go E or W.
Now, think about a big planet that rotates very fast, like Jupiter. These subcells grow in number and shrink in latitude and become narrow bands. The bands on the gas giants are the same as our wind patterns, only there are more of them due to the higher rotation rates and angular momentum. The colors of the bands are due to trace amounts of CH4, P compounds and other colored substrances.
Hard to type a visual description. Hope that helps!