The principle of relativity is not the same as either the theory of special or general relativity. The principle of relativity is much older and forms the basis, along with Maxwell's equations of electromagnetism, for the paradoxes that the theories of relativity resolve.
The principle of relativity is commonly stated as the fact that laws of motion remain the same under any coordinate system of uniform translational motion. Uniform means that the relative velocity of the two coordinate systems is constant. In calculus, using linear algebra, this can be expressed as
--(XA - XB) = C
Where XA is the position of object X in coordinate system A, and XB is the same in coordinate system B. The d/dt means change in what follows relative to time; since we are differentiating position the result is a velocity. C simply represents a constant value, in two or more dimensions a vector, which is their relative velocity. One case of note is where C=0, meaning A and B are not moving relative to each other. However, the two coordinate systems could still be rotated relative to each other, so they are not necessarily the same coordinate system.
A simple example of the principle of relativity is a collision between two balls, A and B. If we center our coordinate system around A we will consider A to be standing still, and B moving relative to A. If we then watch the collision we will arrive at certain physical laws describing the outcome. If we then consider a coordinate system describing the event centered around B, we will see B standing still and A moving relative to B. However, as long as the two coordinate frames are moving with uniform translational motion relative to each other the laws reached to describe the outcome should be the same. This principle is very powerful in practice and is normally used as one of the core postulates in mechanics and the theories of relativity.