For me a **[K3 surface](https://en.wikipedia.org/wiki/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 all but finitely many fibers are smooth curves of genus 1.  I've learned a little about these from here:

* Daniel Huybrechts, [Lectures on K3 surfaces](https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf), Chapter 11: Elliptic K3 surfaces.

Generically a K3 surface admits no elliptic fibration, but among those that do, generically the fiber of $\pi$ is a smooth curve of genus 1 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 singular (i.e. non-smooth) fibers, but with correspondingly worse singularities.  [On  MathOverflow](https://mathoverflow.net/a/87655/2893) there's a nice easy example: the Fermat quartic surface admits an elliptic fibration with 6 singular 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!