Newtonian mechanics does not provide an adequate description of
the natural phenomena when speeds of the order
of 
per second are reached. To be concrete, if you
could go that fast, you could go from Chicago to New-York in
a time of the order of one hundredth
of a second.
Such a speed is not small compared to the speed of light which is
.
The great novelty of the theory of Special Relativity proposed by A. Einstein in 1905 is to use as basic principle that the speed of light is the same for two observers ( technically speaking, we should say inertial observers) moving with respect to each other at constant velocity. This principle is not valid if you are considering the speed of familiar objects which change when you are moving away or toward them.
The implication of the fact that the speed of light is the same
for two observers
moving with respect to each
other at constant velocity is that an interval of time between two events
occurring in a moving frame appears larger for an observer with
respect to who the frame is moving.
This is the celebrated phenomenon of ``time-dilation''.
This allows particles with a lifetime of 
second,
produced 15 kilometers above us to reach the surface of the earth.
If we forget about time dilation, we see that even if this particle
could travel at the speed of light, they could only travel for
600 meters (see Feynman's lectures I-15-4, this means volume I, chapter 15
and section 4)
Another implication of the constancy of the speed of light is that events which are simultaneous in a frame are not necessarily simultaneous in another frame. Since lengths are defined as differences in position at a given time, they will also be frame dependent (see Feynman's I-15-5 and 6).
The changes in lengths and durations for different observers, can be seen
as some sort of ``rotations'' in space-time. In ordinary rotations in
a plane, the
square of the length of a vector having coordinates 
and 
is 
and is left unchanged by a rotation of the
coordinate system about the origin. The length of the vector is then
called an invariant quantity. In special relativity, one can also construct
an invariant out of 
a time interval and a position
interval along the direction of the motion of one observer.
The invariant quantity reads 
(see I-15-7).
General Relativity was developed around 1916 by Einstein. It incorporates gravity according to the equivalence principle: in the presence of sources of gravitational interactions, it is possible, in one point of space-time, to pick a system of coordinates such that the laws of physics appear the same as in the absence of gravity. This is a generalization of the following observation made in the context of Newtonian mechanics: since the gravitational acceleration at a given point is the same for any kind of object, you cannot distinguish between being in an elevator in free fall and being in an elevator moving at constant velocity far away from any source of gravitational interactions.