What was the Law of Gravity better explained by? In mechanics, our professor made the declaration that "all laws of physics" have been disproven. He mentioned several examples including the Law of Gravity, mentioning briefly that it is better explained by Einstein's theory of General Relativity. He also mentioned Bohr's model of the electron, and how it was better explained by other models, which I've since learned. 
Now, haven taken Modern Physics as part of a Physics 3 class, I'm still scratching my head wondering how the theory of general relativity explains attraction between large objects.
Has has Einstein's theory of General Relativity truly explained it or did I misunderstand? If so, how?
 A: I'll answer what I think you're asking: ignore this if I've misunderstood you.

our professor made the declaration that "all laws of physics" have been disproven

This is one of those comments that are true but misleading. To understand why you need to understand what physicists mean by a theory. A theory is just a mathematical model, that is set of equations, that allow us to predict what happens when we feed in a set of initial data.  So we can take Newton's theory of gravity, feed in planet positions, spacecraft velocities, etc and calculate how to send spacecraft to Jupiter - and it works!
But all theories are approximate because they're based on simplifying assumptions. That means no theory is true, but all (good) theories are almost true in a limited range of conditions. Newton's laws work perfectly for sending spacecraft to Jupiter, but we know that they would break down if we attempted to use them to describe relativitic speeds or extremely high densities. To do that we need a more accurate theory, and that's general relativity.
So far general relativity has passed every experimental test we've done, but we expect that general relativity too will break down at very small distances when we expect quantum effects to become important. So general relativity is also only an approximation that we expect to break down when pushed beyond its limits.
So now you see what your professor was on about. There is no physical theory that isn't an approximation, so we expect that every physical theory will have limits beyond which it fails to describe reality. Whether this is a useful point to make outside of philosophy of science classes I'll leave you to decide.
Since you specifically asked about general relativity: in GR you feed in the distribution of masses and the theory tells you how spacetime is curved. Then you use an equation called the geodesic equation that tells you how masses move in curved space. If you do this for e.g. the Earth and the Moon then GR will indeed predict the Moon orbits the Earth (or more precisely they both orbit the barycentre). Whether GR explains why the Moon orbits the Earth depends on exactly what you mean by explain, but this is where we hand over to the philosophers.
A: You probably learned about Einstein's Theory of Special Relativity in your Modern Physics class.  It is Einstein's Theory of General Relativity that provides a more accurate description of what we normally call gravity.  
Basically, general relativity explains gravity not as an interaction between two bodies but as a warp in space-time in the presence of matter.  Rather than thinking of gravity as a force, general relativity treats gravity as a change in space and time itself.  It is more of a theory of geometry than of forces and attractions.  But that is in general relativity, not special relativity.
A: The equations describing "attraction" and "orbital motion" are called geodesic equations
As you can read, the geodesic eqs follow from the field eqs. by specializing the stress-energy tensor. This is a general result for any physical theory described by lagrangians.
A: The truth is that neither Newton nor Einstein actually explained gravitational attraction. They both gave us some equations, but neither gave the explanation concerning the true mechanism of the underlying force. (That's why physics is still so desperately trying to unite GR and QM, and why it cannot get rid of the concept of the graviton.)
Newton simply gave us the equation for a force working at a distance. We can calculate the acceleration an apple is given by the Earth, but we are not explained how the apple knows that several meters away is a much more „massive” (whatever that means) body and what is so attractive about this bigger buddy that our beautiful apple decides to come closer and socialize with him.
Einstein took a shot at it, and he brought the distance much closer, but still failed to explain the very force. Because curvature of space(time) itself simply cannot make an apple move (toward another body or just toward a point). If curvature produces movement, it is only because there is gravitation underneath it. That's why an apple put on the ground rolls down the slope. It takes curvature (of the ground) and also it takes gravitation under (the ground) to do that. If you remove the force and leave only the curvature ... nothing happens to the apple; it will stay right where it was until it experiences real force. So curvature cannot replace gravitational pull, it cannot replace a real force. (Should somebody invoke the Einstein tensor her, please read through first, as I refer to this later on).
You might also learn that making space into space-time changes everything. That adding the time component to space makes things move. And if you still cannot understand how it possibly happens, you will be sent to visualizations showing time is orthogonal to all physical dimensions and told that in space-time we are in constant movement through either one or all of the four dimensions, yet always through time (except for light). So we are actually in constant movement anyway. But how do we know this? Well, nobody is likely to even ask this question, as we are all just so much used to graphs of acceleration (or simple velocity) showing time as orthogonal to the $x$ axis, which makes a beautiful visualization of the changing speed in the form of a curve to which we reply: „Oh, now I get it! Now I know how acceleration works”. Only that when we see a car accelerate on the street, its marks leave no curve. They go just straight as a whistle. So time appears not to be orthogonal to any physical dimension. Therefore the visualization is not real, and we cannot infer any conclusions from it. Because time simply doesn't make things move. Time is rather an indicator of movement. And it is rather movement needed to explain time operationally.
This might be objected to with arguments that the movement is produced by the energy expressed by the Einstein tensor resulting from the stress-energy tensor. Well, I tried to address this issue here: At which point of the universe $R_{\mu \nu}=0$ if there is a source of gravitation (point mass). From Muphrid's answer and the following discussion, you can see that even Einstein's field equations are admitted to be "incomplete", and actually misleading. Textbooks say the equation shows the curvature of space(time) through the Ricci tensor, metric tensor and cosmological constant. If you inquire further, however, you will be told the equation does not actually refer to the space at all, but to matter only ... You will be told that the full space curvature is described by the Riemann tensor consisting of the Ricci tensor (null for the vacuum) and the Weyl tensor. But if you then decide to dig even more and ask the quite obvious question, whether the Weyl tensor is the one responsible for the space curvature that makes massive bodies come together, the answer is ... silence ...
Yet another attempt at explanation is the hypothetical graviton which would „mediate” the gravitational field. However, it still isn't explained how this „mediation” should make things move at all, especially that graviton has no mass to exert any real force on the apple (and how this force would work backwards, against the the direction of the movement of the force „carrier”?).
Obviously, physics is not very eager to admit all that, which is quite 
understandable. It has produced tons of esoteric concepts, maths, fixes, but still no answer as to the basics.
So, going back to the question: Who explained gravity better – Newton or Einstein, the answer is again, neither. To me Newton's version, with all is deficiencies, was at least more honest and did not pretend to have solved the problem it didn't. Because gravity is still action at the distance, and only the Newton's void between two attracting bodies has been now filled with tons of exciting maths that explains exactly nothing.
As a follow-up, I think you should see my answer here: Why Newton's law of universal gravitation is a valid law? What causes any two bodies in the universe attract each other with a force? and watch Feynman's video linked.
To make sure, this is not tell that physics is totally useless and all cheat. All I mean is that physics had better admitted it does not know certain fundamental things, and not just because they cannot be known or that they are metaphysics. Because such a reply means simply evading the answer. And failing to seek answers makes physics fully exposed to complete surprises such as „dark matter”. The miscalculation of the matter-energy of the Universe by some 95% is a clear evidence of its deficits in this area. For the real strength of any science is demonstrated not through developing beautiful multi-level mathematical theories, but through predicting events and phenomena. The deficit of mass-energy of this magnitude is perhaps the best indicator that physics should get interested in the most fundamental questions, such as gravity, yet be fully aware of the fact that mathematical tricks like messenger particles are not going to get it anywhere. Physics needs to decide if it wants to solve problems or just be occupied by them.
