Intuitive

The Minkowski metric encodes the experimental observation that the speed of light has the same value in all inertial frames of reference.

Concrete

\begin{equation} \begin{gathered} (g_{\mu\nu})_{\mu,\nu}=\left( \begin{matrix} 1 & 0 & 0 & 0\cr 0 & -1 & 0 & 0\cr 0 & 0 & -1 & 0\cr 0 & 0 & 0 & -1 \end{matrix} \right) \end{gathered} \end{equation} Thus $x_0=x^0$, $x_j=-x^j$ for $1\le j\le 3$. For real vectors and tensors lowering and raising indices is made by \begin{equation} A_{\mu}^a=g_{\mu\nu}A^{\nu}_a\\ \quad F_{\mu\nu}^a=g_{\mu\alpha}g_{\nu\beta}F^{\alpha\beta}_a\\ \quad \partial_{\mu}=g_{\mu\nu}\partial^{\nu} \end{equation}

Abstract

The Minkowski metric is the correct mathematical tool that tells us how far two events are away from each other in special relativity.

Why is it interesting?