Implicit differentiation is simply the use of the chain rule to differentiate a function. Often this makes it possible to differentiate a function that is difficult or impossible to separate into the form $y = f(x)$.
For example, consider the function $y = e^{xy}$. There is no way to separate $y$, so it is impossible to take an 'explicit' derivative. However, we can find an 'implicit' derivative in terms of $x$ and $y$ using the chain rule.
How do we do this? Well, simply take the derivative of both sides:
$$
\frac{d}{dx} y = \frac{d}{dx} e^{xy}
$$
The left hand side is simply $dy/dx$, or $y'$. The right hand side looks strange to derive, since the equation depends on both y and x. However, a few applications of the chain rule will clear things right up. First, let the 'inside function' be $u = xy$ and the outside function $e^u$. The derivative is, of course,
$$
\frac{d}{dx} e^u = \frac{du}{dx} \frac{d}{du} e^u = \frac{du}{dx} e^u
$$
Now, using the product rule and then the chain rule,
$$
\frac{du}{dx} = \frac{d}{dx} xy = (\frac{d}{dx} x)y + x (\frac{d}{dx} y) = (1)y + x(\frac{dy}{dx} \frac{d}{dy}y) = y + x(\frac{dy}{dx}*1) = y + xy'
$$
Finally, our implicit derivative is
$$
y' = (y + xy')e^{xy}
$$
and we can easily solve for $y'$.
So you see that implicit differentiation is the result of giving up on separating variables to get a derivative in terms of $x$ only, and cleverly using the chain rule to instead get a derivative in terms of $x$ and $y$.