What happens if the gravitational force on a satellite is greater than the centripetal force needed to keep it in circular motion? Will the satellite continue to orbit the Earth, albeit not in a circle? Why? I would greatly appreciate it if someone could help me address this query. :) Also, I think someone asked a similar question sometime ago and one of the answers said that the satellite will travel around the earth in an elliptical orbit, but I'm not sure why. 
 A: Assuming the satellite is far enough away from the Earth that the atmosphere doesn't matter:
The centripetal force required to keep something moving in a circle depends on two things: the radius of the circle and the speed of the moving object. The bigger the radius, the less force is needed; likewise, for faster speeds, more force is needed.
If the gravitational force is greater than the required centripetal force, then the satellite will begin to be pulled inward; if, on the other hand, the gravitational force is less than the centripetal force, then the satellite will begin to fly outward.
So if we start in a position where gravity is greater than the centripetal force, then the satellite will begin to move inward. As it does so, it picks up speed; therefore, the required centripetal force becomes greater and greater, until it eventually is equal to gravity. At this distance and speed, if the satellite was moving completely laterally (i.e. not moving inward or outward), a circular orbit could be sustained. But the satellite is not moving completely laterally here; due to the fact that it was pulled inward by gravity, it's still moving inward, so it overshoots this "equivalent circular orbit" and keeps on moving inward. But now gravity is less than the required centripetal force, and so the satellite's inward motion starts to decelerate, until its inward motion completely stops and it starts flying outward again. By the same logic, it overshoots its "equivalent circular orbit" again, flying outward past it. This process of oscillating about the "equivalent circular orbit" repeats indefinitely.
Now that we've demonstrated that the satellite's distance from Earth oscillates, it should make sense that it would travel in an ellipse (which is a shape which, as you travel around it, has a distance from its focus that also oscillates). It's actually somewhat nontrivial to prove that the shape of a gravitational orbit is an ellipse; that has to do with the particular character of the gravitational force, and the calculus is somewhat involved (you can find the relevant proof in most classical mechanics textbooks). But this should give you an intuitive idea of why an elliptical orbit would make sense.
A: I think you are confusing centripetal force with centrifugal force. Centripetal force is the gravity force projected along the instantaneous radius of the orbit of planet. I a circular orbit it is the same as force of gravity.
Centrifugal force is the force due to inertia of the satellite and will try to pull it out and fight gravity. Its magnitude is the same as centripetal but with reverse direction. $$ F= \frac {  m_{satellite} \times v^2}{r}$$ It is a fictitious force which must be added to correct the balance of forces in an inertial frame rotating with the satellite. 
If this centrifugal force is less than the gravity at a given point the stellate will descend into lower orbit while accelerating, because of the additional pull of gravity which is gradually stronger at lower orbits. As it accelerates the centrifugal force will increase as well and the satellite will start to climb to higher orbit.
This will lead to a elliptical orbit, which is the typical orbit of solar system planets .
Usually in planetary mechanics they just use Newton's gravity force which is more direct and exact vector:$$ F = -GMm/r^2 \times (unit\space  vector\space of R)  $$ negative sign indicates direction of the force is in to the earth.
In certain cases and depending on the angle and velocity of descent, the stellate in descending to lower orbits, may pick up so much velocity that it will go around the earth while having taken advantage of the great earth velocity and leave the other side and escape earth's gravity. They call this gravitational assist and NASA uses it as a way to send objects far away into space and save on fuel.
