Changing Electric Field causes Magnetic filed and changing Magnetic Field causes Electric Field. Is there anything similar in relation to Gravitational Field? What sort of field is created by varying Gravitational Field?
The only way this would work is to have a set of equations describing gravitation that are structurally the same as Maxwell's equations. Then you can have something similar to the concept of induction and induced fields. Ted Bunn above has pointed out already that gravitomagnetism is the name for this setup. Check out the wikipedia page on this:
It has a nice summary of the GEM equations and puts them right up next to Maxwell's equations so you can see the equivalences.
The answer is very much yes! In fact, Gravity Probe B detected frame dragging effects by comparing rotation speed of four extremely precise gyroscopes onboard. As one might think, the force is very small for an object like earth rotating so slowly. However for fast rotating massive bodies, is has noticeable effects. Supermassive black holes in active nuclei have enormous jets that are very closely aligned even at megaparsec (millions of light years) distances. As far as the nature of the field created, it is very much congruent to the relationship between the electric field and the magnetic field.
Isn't frame dragging an example of this?