Did Einstein completely prove Newton wrong? If so, why we apply Newtonian mechanics even today? Because Newton said that time is absolute and Einstein suggested it relative?
So, if fundamentals are conflicting, how can both of them be true at a time?
In physics, it is often true that theories or theoretical paradigms with vastly, "qualitatively" different assumptions and "pictures to imagine what is going on" yield virtually indistinguishable predictions, and Newton's vs Einstein's physics is the simplest example of that.
According to Newton, for example, time was absolute. According to Einstein, time depends on the observer but time $t'$ according to one observer is expressed as a function of time $t$ of another observer as $$ t' = \frac{t - \vec v\cdot \vec x/c^2}{\sqrt{1-v^2/c^2}}\sim t - \frac{\vec v\cdot \vec x}{c^2}$$ This approximation is good at low enough velocities, $v\ll c$. You may see that the "times" only differ by a small number that depends on $1/c^2$ which is $10^{-17}$ in SI units (squaread seconds over squared meters). They're different in principle but the difference is so small for achievable speeds that it is (almost) unmeasurable in practice.
Similar comments apply to many other phenomena and deviations. Newton would say that they're "strictly zero"; Einstein says that they are "nonzero" but their size is tiny, comparable to $1/c^2$ times a "finite" expression.
Analogous comments apply to classical physics vs quantum mechanics. Classical physics often says that something is strictly impossible, some quantities are zero, and so on. Quantum mechanics says that they are possible, nonzero, etc. but their numerical size is $\hbar$ times a "finite expression" which is again unmeasurably tiny for macroscopic objects.
In both cases and others, one may prove that the $1/c\to 0$ or $\hbar\to 0$ limit of the more complete theory is exactly equivalent to the older theory. So (special or general) relativistic physics reduces to Newton's physics in the $c\to \infty$ limit, for example.
Newton's laws of motion and his law of gravitation are still around and taught because they predict very well the real world under normal circumstances; when things aren't too fast or their gravitational field isn't too strong. This is why it took more than 200 years for more complete theories to arise.
Additionally, Newton's laws are relatively simple when compared to Einstein's theories of relativity. If a more complete theory of motion and gravity were put forth by someone, and it were just as simple as Newton's laws, I'd expect Newton's laws to eventually only appear in history texts and physics footnotes.
In short, Newton's laws is around mainly due to its accuracy for everyday concerns and its simplicity. Did Einstein prove Newton wrong? Yes, I suppose so, but do remember that wrong doesn't mean not accurate for our purposes. More importantly, it did not make Newton's work obsolete.
I'd like to add to Lubos's Excellent Answer but perhaps to downplay the difference between Newton/ Galileo and Einstein a little. As a principle, relativity was embraced every bit as fully by Newton and Galileo as it was by Einstein - it's just that Einstein had a few more experimental results he had to gather into relativistic thinking.
The whole point of special relativity is that velocity is a relative concept insofar that the physical laws will seem the same to all inertial observers.
This concept was well appreciated by Galileo and Newton. See for example the quote of Galileo's character Salviati in the Galileo's Ship Thought Experiment of 1632. Saviati's narrative is clearly saying that there is no experiment whereby one could tell whether or not the ship were moving uniformly.
The difference between Einstein and Newton's thoughts about relativity is actually quite small: the key difference is the assumption of whether or not time and simulteneity could be relative. Newtonian / Galilean relativity is simply the unique relativity defined by Salviati's relativity principles given an assumption of absolute time. Einstein's STR relaxes that assumtion, but still applies the exact same relativity postulates as stated by Salviati. With the relaxed assumption, the transformation laws between inertial frames are no longer uniquely defined by Salviati's narrative, but instead there are a whole family of relativities, each characterised by a unversal parameter $c$ which are consistent with Salviati's words. Galilean relativity is the member of the family with $c\to\infty$. So now there is a parameter $c$ which we must experimentally measure: and the Michelson-Morley experiment showed that light speed transformed in exactly the way that a Salviati-compatible relativity with a finite $c$ foretold: i.e. it is the same for all inertial observers. See here for more details, as well as the Wiki links I give in that answer.
I've therefore never felt the difference between Newton and Einstein on STR are particularly big. But then I was born in the 20th century: to someone of Newton's deeply religious era, with strong Unitarian church faith like Newton, a relative time would be a huge difference, and he had no experimental grounds to doubt it.
As for gravitation, this is where Newton's and Einstein's ideas are very much a different paradigm: in the former, things exert forces on one another across the voids of empty space, whereas the latter is wholly local and geometrical in nature. In the second paradigm, things moving through spacetime respond to spacetime's local, and unhomogeneous, properties: "massive" things distort the properties of spacetime, which then acts locally on things moving through it, so the "action at a distance" and forcelike character of gravitation is dispensed with. Moreover, astronomers witness more and more phenomena which are altogether at odds with Newtonian gravity: the differences are no longer small quantitative ones, but totally qualitative. See for example the gravity wave begotten spin-down of the Hulse-Taylor binary system.
Having said this, don't believe for a moment Newton was altogether happy with his theory. Its action at a distance nature was something he was unhappy with and, would he have had the 19th century geometrical tools that let us write down GTR's description, he might have made considerable progress in dispensing with it. The idea of space having a non-Eucliean geometry induced by the "matter" within it was an idea explored by Gauss, Riemann, Clifford and others. See for example my exposition here.
So, in summary:
The special theory of relativity builds on and makes relatively minor changes to what was a near complete edifice built by Newton and Galileo. They key generalisation is precisely the relaxation of the assumption of absolute time.
The general theory of relativity pretty much fully replaces Newton's paradigm. It is a wholly different way of thinking. However, (1) Einstein did look to Poisson's equation (the equation for gravitational potential inside a system of distributed masses in the Newtonian theory) for hints on things like the order of derivatives that must be present in a description of gravity: see this most excellent summary here by Eduardo Guerras Valera of Einstein's "The Meaning Of Relativity" downloadable from Project Gutenberg and (2) he calibrated his field equations by comparing their low gravitation limit to the Newtonian theory. The scale factor on the RHS in the field equations $R_{\mu,\,\nu} = 8\,\pi\,G \,T_{\mu,\,\nu} / c^4$ (in the SI system: natural Planck units set $R_{\mu,\,\nu} = 8\,\pi\, T_{\mu,\,\nu}$) is uniquely defined by this requirement.
Einstein extended the rules of Newton for high speeds. For applications of mechanics at low speeds, Newtonian ideas are almost equal to reality. That is the reason we use Newtonian mechanics in practice at low speeds.
On a conceptual level, Einstein did prove Newtonian ideas quite wrong in some cases, e.g. the relativity of simultaneity. But again, in calculations, Newtonian ideas give pretty close to correct answer in low-speed regimes. So, the numerical validity of Newtonian laws in those regimes is something that no one can ever prove completely wrong - because they have been proven correct experimentally to a good approximation.
Both Newton and Einstein are correct. However Einstein is more correct. Physical laws are in fact models made up by us that allow us to describe certain phenomena.
Newton's laws allow us to describe certain situations in a very accurate and simple way, but fails in other situations. You should actually interpret Newton's model as a very good approximation to many experiences.
Einstein's model is a more precise model. It covers Newton's model and the situations where it failed, especially for high velocities.
Did Einstein completely prove Newton wrong? No - the theories are based on slightly different first principles when describing force.
From Newton's Principles definition #4 in 1687:
Impressed Force - This force conflicts in the action only; and remains no longer in the body when the action is over.
In Einstein's second paper on relativity in 1905:
Radiation carries inertia between emitting and absorbing bodies.
In Einstein's world, he explicitly concludes that not only does something receive a "kick" from the momentum of the energy, but the internal inertia (i.e., the inertial mass) of the body is actually increased. This is clearly different from Newtons definition #4.
No current theory is complete. Pick your boundary conditions and scale of operation for a "good enough" convenient answer.
Lightspeed, c, enforces maximum information transfer rate. Newton's constant, Big G, scales gravitation. Planck's constant, h, enforces uncertainty in measurement; h-bar is the fundamental unit of action. Boltzmann's constant, k, unites classical thermodynamics with statistical mechanics. Thermodynamics plus the Beckenstein bound (equilibrium of black holes) obtains GR. Setting k = 0 gives conservative mechanics; k > 0 obtains corrections to the geodesic equation of motion: thermodynamics and dissipative processes that produce entropy.
? h=h G=G c=infinity mechanics, electrostatics: h=zero G=zero c=infinity classical physics: h=zero G=G c=infinity quantum mechanics: h=h G=zero c=infinity special relativity: h=zero G=zero c=c general relativity: h=zero G=G c=c quantum field theory: h=h G=zero c=c quantum gravitation: h=h G=G c=c (non-predictive)
Notice one thing; Neither Newton nor Einstein actually explained the mechanics. They stated their equations differently, but none of them explained the force. (I'm going to get a lot of upvotes for this, but ...)
For Newton gravitation was just a force at a distance. He didn't even try to show the mechanics behind the attractive force of gravity.
Einstein introduced curvature to replace the force, but curvature can't explain the onset of motion if there is no gravitation under the curvature. Even if one introduces the concept of space-time and says that we constantly "travel" through time - it does not explain how movement through time converts into movement along some spatial axis. You need some additional force to make a push (pull) and change the "direction" of movement, just like you need some additional force to change a movement along y-axis to a movement along x-axis. (Also, this "time travel" converting into space travel is showed through diagrams where time is orthogonal to space axes; in reality, time is not orthogonal to space, which is proved by the fact that an acceleration does not need to produce a curved movement, which it always does on paper diagrams, because of the way time is drawn there).
And by the way, Einstein used the term "proper time" which can be understood that locally time is absolute. That's commonly admitted by physics now when we say that the moving observer will not notice he is undergoing time dilatation (without comparing with "stationary" frame of reference).
So Einstein did not prove Newton wrong, because Newton made no claims about the mechanics of gravity, and Einstein failed at explaining them.
Did Einstein completely prove Newton wrong?
Rather, Einstein recognized where Newton had (largely) been "not even wrong".
Einstein demanded and (partially) provided definitions of notions, especially relating to geometry or kinematics, such as how to measure "straightness", "distances", "speeds", "curvature"; which Newton had used (or relied on being "in some sense negligible") without any general definition, but at best only by appeal to some parochial artefacts (such as particularities of "our solar system").
Because Newton said that time is absolute and Einstein suggested it relative?
Well -- along the way, Einstein recognized that a definition of how to measure "simultaneity" is necessary; and he provided his famous concrete definition. Newton, on the other hand, certainly had not provided any actual solution to the problem; which makes doubtful that he had even recognized the problem at all.
If so, why we apply Newtonian mechanics even today?
We may apply certain formulas and related mathematical methods which had been established within Newtonian mechanics today only as far as formulas of Einsteinian mechanics could be made to mimick them; for instance in suitable circumstances involving speeds much smaller than $c$, or curvature only as small as may be found in "our solar system".
An advantage is that the relevant formulas tend to be mathematically simpler than those of "full blown RT". However, in any case we certainly rely on the foundation laid out by Einstein.
