How do we know that Earth is not perfect sphere? I've just read here that:

Equatorial radius = 6378.16 kilometers. Polar radius = 6356.78
  kilometers, so the difference in circumference is 71.1 kilometers. It
  is not a perfect sphere, but kind of pear-shaped.

How correct is that information and what exactly are Equatorial radius and Polar radius with diagram if possible?
 A: The shape of the earth is called a Geoid.

Read here: http://www.esri.com/news/arcuser/0703/geoid1of3.html
A: Most interesting question, since it can be discussed from several different points of view which are fascinating in themselves.
If you know actual shape of Earth that you need only to measure parameters which define the geometrical figure. For perfect sphere there is the only parameter - the radius. In principle you can drill a hole into Earth and measure how deep it is, but neglecting the practical difficulties (the deepest drill has some 12 kilometers) how can you be sure you hit the center of the sphere? In practice you would measure circumference of the globe and then compute the radius. Less direct approach (chosen by Eratoshtenes and others) is to measure part of a meridian. This brings another problem - how you know which part of the entire circumference you measured, solved by measurements of geographical latitude. This relies on astronomical measurements - in principle you measure distance of stars from zenith. Since zenith is defined by direction of local gravitational force the results depend also on distribution of mass in the sphere, not only on its shape.
If you release the condition of the perfect shere the next shape is a flattened ellipsoid (Earth is reasonably supposed to be flattened by its rotation). Than you don't have polar or equatorial radius but semiaxes but still you can go along a meridian and the equator and measure their length or length of their parts.
In fact the shape of Earth is more complicated. General shape of the solid surface (or equipotential surfaces given by distribution of mass and amount of rotation) can be derived from observation made by external observer (a satellite) or - in principle - by measurements made only within the 2D surface along recipes given by Gauss and Riemann in other context. Strictly speaking (to a non euclidean geometer) there are no parallels on Earth. In fact I am little confused that non euclidean geometry - given that spherical Earth was on our mental eyes so long - was first discovered for hyperbolic surfaces. 
A: There is a precession of the earth's axis of rotation -evidenced by astronomical observation- that is most easily explained by Newton's laws and a non-spherical earth.
@Ross Milikan suggested in a comment that this was not an answer because precession only shows that the moments about the principle axes are unequal - and this could be caused by a non-uniform distribution of mass inside a spherical earth.
That's not correct, in fact precession measurements (of orbiting satellites and the earth itself) provide our most accurate estimates of the figure of the earth. The distribution of density in the earth is constrained by seismological evidence, and support a generally 'onion skin' earth structure. Significant differences in uncompensated density distribution would give rise to non-hydrostatic stresses that would exceed the observational and experimental known strength of earth materials.  Convection currents inside the earth give rise to non-hydrostatic stresses, but the amount of convection (to account for the entire moment) would be too vigorous to be consistent with observed geothermal heat flow. When we consider all the geophysical evidence available - the best explanation for J2 not being zero is that the shape of the earth is ellipsoidal and not a simple sphere.
Geophysics often arrives at it's best answers by searching for a model that can fit all the available information - gravity, heat flow, astronomical, and materials science. All real data sets are incomplete, and geophysicists are often trying to solve the 'inverse problem' anyway.  More accurate answers are obtained by using a variety of independent data, than by using only one imperfect set.
Combining precession data and other geophysics data to calculate a physical model with which to estimate the polar and equatorial radii of earth is no less valid than measuring these with a transit and level (which is a practical impossibility anyway.)   
A: As to, how do we know?
Originally we sent teams of (mostly French) madmen to measure the distance on the ground between two distant mountain tops near the equator, in the jungles of south America, and then the same thing between places in the arctic.
They then measured the angular distance between the points by making astronomical measurements of star positions. By comparing the circumference (ground distance) and the arc (angle distance) you get the radius of the Earth at that point.
These expeditions took many years and generally because of the difficulties of making the precise distance measurements - were inconclusive.  Now we just use gps and radio telescopes to make amazingly accurate geoid maps.
