Einstein gravity versus Newton's gravity What's the basic difference between the gravity as seen by Einstein, and that by Newton? 
 A: 
What's the basic difference between the gravity as seen by Einstein,
  and that by Newton?

Newtonian gravity is an instantaneous force, i.e., action at a distance, coupled to gravitational mass (conceptually different from inertial mass).
General Relativity is a local theory (no action at a distance).  Einsteinian gravity is the curvature of spacetime and the coupling is between mass-energy and geometry; "matter tells spacetime how to curve, spacetime tells matter how to move".
A: In at least one basic respect, both general relativity (gravity according to Einstein) and Newtonian gravity are similar; both describe gravity as a gravitational field on a space.  In other words, they are both classical field theories.  
In the case of general relativity, that field is a pseudo-Riemannian metric $g_{\mu\nu}$ on the space, and the space is a 4-dimensional topological space called spacetime, while in the case of Newtonian gravity, the field is a vector field (if you are describing it by the gravitational acceleration $\mathbf g$) or a scalar field (if you are describing it by the gravitational potential $\Phi$) on three-dimensional Euclidean space.
In the case of general relativity, the gravitational field tells you the geometry of spacetime, and it's the curvature of this geometry that particles "ineract with" in when they move around.  The gravitational field is determined by the energy-momentum content of spacetime through Einstein's Equations.
In the case of Newtonian gravity, the gravitational field tells you the acceleration that a particle would feel at any given point in space, but in contrast to general relativity, the geometry of the space itself is not altered by the sources of gravity (masses in this case).
In a highly simplified nutshell:
General relativity describes gravity as spacetime curvature while Newtonian gravity describes it as something living on top of a static space with no curvature.
A: Often people get confused by the additional complication that Newtonian and Einsteinian gravity are often discussed in different mathematical formalisms. This can tend to obscure the physical differences. If you are game for the mathematics then Misner, Thorne and Wheeler (check it out of a library or get it second hand unless you are really serious about this business) has a wonderful chapter which puts both theories side by side in the same language (differential geometry). The key difference is that Newtonian gravity has a privileged separation of spacetime into space and time, whereas Einsteinian gravity just has spacetime.

Edit: to be absolutely clear, Newtonian gravity can be written as spacetime curvature! This is counter to the common statements about the novel thing in GR. The key difference is that Newtonian gravity has extra absolute structures that GR does not have: absolute time and space, a preferred separation of spacetime into time and spatial parts, absolute simultaneity, and a curved connection that is not the special one derived from a spacetime metric (Christoffel).
In mathematical form:
$$ \begin{array}{ll}
R_{00} = 4\pi\rho;\text{all others vanish},& \ \text{Newtonian} \\
R_{\mu\nu}-\frac{1}{2} g_{\mu\nu} R = 8\pi G T_{\mu\nu}, & \ \text{Einsteinian}
\end{array}$$
with a few other relations I've not written (see MTW chapter 12 for details).
A consequence of the formalism is that the Newtonian equation is a constraint equation - it does not describe a propagating degree of freedom. No gravitational waves, gravitons etc. No speed of light limit for gravity. All matter has an instantaneous gravitational effect on all other matter. This is different in GR since the field equation is a wave equation which describes the propagation of gravitational disturbances from one point to another at the speed of light.
What GR has that Newton does not is a spacetime metric of Lorentzian signature. This metric has a privileged role in that all other structures (connections, curvatures, etc.) are derived from it. There is essentially nothing else to Einstein gravity. That is why it is so elegant in the geometrical formalism. This metric actually comes from special relativity. But the metric was a fixed structure in SR, almost similar to the absolute time and space of Newton (don't tell anyone I said this). The new thing in general relativity is that Einstein lets the metric "flap around" so to speak - to change from place to place and time to time in response to what matter is doing.
A: One more aspect. Roughly speaking, in General Relativity "energy" is attracted $(E/c^2)$, while in  Newtonian gravity - only mass. And there is no time dilation in  Newtonian gravity. 
A: In Newtonian Gravity, space is like 3-dimensional graph paper, and objects are moving through space at an absolute time.  The objects path curves because a force is present. Without that force (gravity) objects will continue in a straight line.
Where Newton's idea of Gravity is 3D space with Time a constant, Einstein conceived of 4D space -- called space-time. In this structure, time is not absolute but a dimension or variable in the structure such that (x,y,z,t) exist for a given event.  Objects moving through space-time curve not because they are "pulled" by the force of gravity, but because they are taking the shortest distance through curved space-time.  
Amazingly, for Einstein, you can have curved space-time without matter so an object might start moving in curves even if nothing is present.
A: Newton saw the universe, as a 17th century man would, as having a fixed time speed everywhere and lengths that appeared the same to all observers.  Within this model, he created a really excellent set of rules for motion under the influence of gravity.
Einstein allowed for the speed of time and length measurements to vary based on the observer's reference frame.  The resulting complications can be addressed using 19th century mathematics (Einstein famously said to a mathematician friend, "Grossman, you must help me or I will go mad!").  However, I think people go too far when they turn around and "see" the mathematics as defining a sort of higher reality. I think it is just a model; a very wonderful model that works, but still just a model to address the differences in viewpoints of time and length measurements.
If one takes a more prosaic view, and simply says "Newton was pretty much right other than adjusting for different times speeds and measured lengths," then one can develop a parallel model for motion under the influence of gravity, that works every bit as well at matching observations including gravitational lensing, the Shapiro time delay, planetary precession, satellite precession, relativistic rocket travel, photon sphere, gravitational waves, and the dynamics of falling into a non-roating black holes. The key is to retain all of Newton's laws, include conservation of energy, but allow the local traveler to experience motion according to his reference frame (he feels he is closer to the gravitational source, and that path lengths are shorter, and that the distant viewer has a clock that runs slower due to velocity and faster due to less gravity). For example, remove Mercury and put a spaceship in its place with a physicist and perfect measuring equipment - at no point in the orbit would he ever measure an error in Newton's laws nor see a day's motion that was inconsistent with Newton's laws, but sure enough, after 1 Mercury year, he'd find that he ended up with a precessed perihelion.
Conversely, there are things that the geometric curved spacetime models struggle to explain.  For example, as people note, it is vital that there not be an absolute maximum time speed (which would imply an absolute zero velocity).  Yet I can populate the universe with clocks created at the big bang and set in a no-dipole-in-the-cosmic-microwave-background-radiation reference frame, allowing them to separate with universal expansion.  It is impossible to create a clock in any other reference frame that could be beside one of my clocks and record more time having passed since creation.  Hence it appears that I have defined a maximum time rate and an absolute zero velocity.  (In fact, it is necessary for us to define an absolute zero velocity in order to define an age of the universe.)
There are other problems with the geometric mathematical solution, in that the complexity leads to false conclusions or the reliance on adages that may not be true in all cases.  For example, one response says that "Objects moving through space-time curve not because they are "pulled" by the force of gravity, but because they are taking the shortest distance through curved space-time."  But consider the path of two clocks starting at a point on the moon and returning to the origin.  One clock is thrown vertically upward, while the other is thrown very fast to circle the whole moon at a very low altitude and return.  The clock thrown vertically will record more time than a clock that sits at the origin, while the clock that flew all around the moon will record less time than the clock at the origin.  The vertically thrown clock followed a "shorter distance" through spacetime than the still clock, but the low angle throw followed a longer distance through space time than the still clock, yet both paths are geodesics.
And there are more problems, like the conclusion that the astronaut twin ages less because she accelerated and later reversed direction while her brother stayed still.  But have the astronaut accelerate to leave Earth on January 1 at noon, and have her stay in position fixed relative to the stars, and her brother will come around next January 1st to find that his sister has aged more than him by a fraction of a second (this is true even if the circling the sun twin is on board a spaceship with no gravity rather than on the planet Earth).
That is - Newton is not so bad.  We can adjust Newton to account for changing time speeds and lengths, and do a fine job at dealing with motion through the universe under the impact of gravity using his laws.  The conventional curved space model is useful, but it is not proven to have any more "truth" than Newton's model with adjustments, and it can lead to incorrect conclusions.
To anyone who feels angry at this response, please don't down vote unless you can address the three specifics I mention - an apparent universal maximum time speed, the tossed clocks whose paths can be either shorter or longer distances through spacetime than a still clock, and the twin paradox problem's resolution having nothing to do with who accelerated or who changed direction. And if you can explain those things, I will be very pleased to learn from you.
