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Some good introductory info here:

http://en.wikipedia.org/wiki/Magnetosphere

 

Short answer, the strength is determined by how fast the planet is rotating compared with its size (ala size * speed = strength). Some planets and moons have them, others do not. The case of Mars is particularly interesting for its lack of one, and what that might mean regarding our own's future, since it apparently had one at some point in its distant past (its loss may be a factor in its loss of atmosphere, since the field is thought to reduce atmospheric loss due to movement through space).

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All the planets have some magnetic field. The generate process is not fully understood and is a topic of current research.

 

A good example is Jupiter. It has a very strong magnetic field (20,000 times stronger than Earth's), but no one knows why. More specifically, no one knows the generation mechanism. Also, the magnetic fields of Jupiter interact with it's moons, most notably Io. Large currents flow between them.

 

Similarly the other gas giants Saturn, Uranus and Neptune have large magnetic fields. Again, it is not really understood how these fields are generated.

 

It was thought that small rocky planets would have no (or little) magnetic fields. This is true for Venus, but not for Mercury. Presumably, Mercury has a metallic molten core which due to the dynamo effect produces the field.

 

The Moon and Mars have very weak fields, but rocks on their surfaces suggest that they did once have stronger fields. They probably generated their fields in via a molten core and the dynamo effect.

 

The Earth's magnetic field is also attributed to the dynamo effect.

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