How do derivatives relate to limits?

The derivative of a function $f \left(x\right)$ at a point ${x}_{0}$ is a limit: it's the limit of the difference quotient at $x = {x}_{0}$, as the increment $h = x - {x}_{0}$ of the independent variable $x$ approaches $0$. In mathematical words:

$f ' \left({x}_{0}\right) = {\lim}_{h \to 0} \frac{f \left({x}_{0} + h\right) - f \left({x}_{0}\right)}{h}$

The definition can also be stated in terms of $x$ approaching ${x}_{0}$:

$f ' \left({x}_{0}\right) = {\lim}_{x \to {x}_{0}} \frac{f \left(x\right) - f \left({x}_{0}\right)}{x - {x}_{0}}$

So derivatives are special limits, which help in getting useful information about functions and their behavior.

Historically, Newton (XVII century) was the "inventor" of derivatives (together with Leibnitz: there's a complex debate between mathematicians about the paternity of the derivative). He introduced this new concept without formalizing explicitly the definition of limit. Unfortunately, the theory of derivatives is not well grounded without a precise notion of infinity and infinitesimal. So, to fix the foundations of analysis (in particular differential and integral calculus), Cauchy and Weierstrass (XIX century) developed the notion of limit.
In 1960s, some mathematicians worked out an alternative foundation of infinitesimal calculus, based on the original ideas of Leibniz and without limits. It's called non-standard analysis, and has been very discussed in the last decades.