I may add to the above already-excellent question, that the reasoning behind the introduction of magnetic field can be explained a little less mathematical.

Basically, first thing to consider is an electric charge. When it does not move in our reference frame it does not produce any magnetic field, just electric one. Now, if it starts to move, we percieve its width (and width of spacetime) parallel to its moving direction to contract so that for every $dl$ we have $dl' = \sqrt{1-v^2} dl$ - Lorentz contraction. Now, the field lines of electric field also contract - and the closer field lines are - the stronger the field. This added influence is simply called 'magnetic field' but it is not a separate entity. So, magnetic field is a relativistic shadow of electric field. This is reflected in Maxwell equation, where charge density only apperas with the electric field, not the magnetic one. So, charge causes electric field and our relative movement w.r.t. charge causes our perception of this field to change.

This, however, is only half of the truth. This is because changes in electric fields have been found to propagate in vacuum - these are electromagnetic waves. In vacuum, Maxwell equations become completely symmetric - and one can not distinguish cause and effect when it comes to electric and magnetic fields. This is where 'relativistic reasoning' has to take precedence over our classic intuition and we have to accept that those fields - however distinct they might seem - are just one entity - the electromagnetic field.

I may add to the above already-excellent answer, that the reasoning behind the introduction of magnetic field can be explained a little less mathematical.

Basically, first thing to consider is an electric charge. When it does not move in our reference frame it does not produce any magnetic field, just electric one. Now, if it starts to move, we percieve its width (and width of spacetime) parallel to its moving direction to contract so that for every $dl$ we have $dl' = \sqrt{1-v^2} dl$ - Lorentz contraction. Now, the field lines of electric field also contract - and the closer field lines are - the stronger the field. This added influence is simply called 'magnetic field' but it is not a separate entity. So, magnetic field is a relativistic shadow of electric field. This is reflected in Maxwell equation, where charge density only apperas with the electric field, not the magnetic one. So, charge causes electric field and our relative movement w.r.t. charge causes our perception of this field to change.

This, however, is only half of the truth. This is because changes in electric fields have been found to propagate in vacuum - these are electromagnetic waves. In vacuum, Maxwell equations become completely symmetric - and one can not distinguish cause and effect when it comes to electric and magnetic fields. This is where 'relativistic reasoning' has to take precedence over our classic intuition and we have to accept that those fields - however distinct they might seem - are just one entity - the electromagnetic field.