Did relativity make Newtonian mechanics obsolete?
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.