How important are inequalities?

Inequalities are extremely useful in mathematics, especially when we deal with quantities that we do not know exactly what they equate too. For example, let $p_n$ be the $n$-th prime number. We have no nice formula for $p_n$. However, we do know that $p_n \leq 2^n$. Often, one can solve a mathematical problem, by estimating an answer, rather than writing down exactly what it is. This is one way inequalities are very useful.

There are a lot of inequalities in mathematics that are more or less important, for a list you can see here.

It is not simple to establish a rank of importance for them, but I think that the most important is the triangle inequality. In the simplest form this inequality states that, for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This captures a fundamental character of the notion of distance that agree with our intuition in the euclidean space. But it can be generalized to more abstract spaces (as the spaces of functions in functional analysis) so that, also in these spaces we can define a notion of distance.

To prove the triangle inequality in these spaces we need some other inequalities and the more relevant are the Holder and Minkowski inequalities that are used to prove when in a vector space we can define a norm, and, from this norm, a distance.

I have a feeling that what you actually seek are examples of "famous" inequalities being put into good use and not just the notion of inequality as a general attribute.

For, if we stick to the notion of inequality in general, a prime example of why that is essential is in defining the real line.
Dedekind cuts that define the reals are partitions of the ordered field $\Bbb{Q}$. If you do not have an ordering (inequality relationships) amongst it's members, you cannot define $\Bbb{R}$ in this way.

For an example of an inequality as a "formula" , consider the $LM$-inequality, used in complex analysis, that gives an upper bound for a contour integral, thus having a variety of applications.
If f is a complex-valued, continuous function on the contour $\Gamma$, the arc length of $\Gamma$ is $l(\Gamma)=L$ and the absolute value of $f$, $|f(z)|$ is bounded by a constant $M$ $\forall$ $z$ on $\Gamma$, then it holds that $$\int_\Gamma|f(z)|dz\leq ML$$