For me a K3 surface will be a smooth complex projective variety of dimension 2 that is simply-connected and has trivial canonical bundle. Given a K3 surface $X$, an elliptic fibration $\pi \colon X \to \mathbb{C}P^1$ is a proper morphism with connected fibers such that almost all the fibers are smooth curves of genus 1. I've learned a little about these from here:

• Daniel Huybrechts, Lectures on K3 surfaces, Chapter 11: Elliptic K3 surfaces.

Generically a K3 surface admits no elliptic fibration, but among those that do, generically the fiber of $\pi$ is smooth at all but 24 points, where the fiber is a rational curve with a single double point.

Huybrecht also catalogues the less generic cases where $\pi$ has fewer non-smooth fibers, but with correspondingly worse singularities. On MathOverflow there's a nice easy example: the Fermat quartic surface admits an elliptic fibration with 6 singular (i.e. non-smooth) fibers, each of which has 4 double points.

But I'd like to see concrete easy examples of elliptic fibrations with 24 singular fibers.

By 'concrete easy examples' I mean that ideally I would like there to be simple explicit formulas for the K3 surface $X$, the hyper-Kähler structure on $X$, the elliptic fibration $\pi$, the 24 points on $\mathbb{C}\mathrm{P}^1$ with singular fibers, the double points on these fibers, and also the points of $X$ where $d\pi$ is not injective. But of course I'll settle for whatever I can get!