The Magnetosphere is the region of space surrounding a celestial object that is affected by the objects magnetic field.
The Earth's magnetosphere is thought to be generated by its rotating iron core. This is commonly referred to as a Dynamo. In order for a planetary core to act as as a dynamo it must contain a rotating liquid metal and there must be convection. Current theory is that the core of Mars has cooled enough to solidify or become plastic, but seismic investigations will be required to confirm or disprove this conclusion.
A magnetosphere of sufficient strength will serve to protect the occupants of a planet from the harmful solar wind and radiation of their star and the surrounding cosmos. The results of this protection can be seen as aurorae in the far northern and southern latitudes.
Because Mars lacks a significant magnetic field, its magnetosphere offers negligible protection from solar wind and radiation.