[Étale cohomology](https://en.wikipedia.org/wiki/%C3%89tale_cohomology)
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Not really an ELI5 concept since to have any need to even know such a thing requires specialization in number theory and/or algebraic geometry which are highly sophisticated fields of study.
The main idea, however, is to take a hint from topology when trying to conceptualize geometry on objects which are not apparently geometric. Topology is made from open sets which we usually think of as subsets of the space. But, instead, we can think of them as an immersion from one set into another. That is, if U is a subset of X, then we can think of U as the arrow U -> X. If we think in terms of arrows rather than subsets, then we can redefine topology as a property of a collection of arrows rather than a property of subsets. This means that if this particular arrow structure appears somewhere else then we can think about it in terms of topology even if it has nothing to do with topology.
In topology, the study of continuous functions is super important and this is done by understanding sets of functions with different properties on different open sets. These sets of functions have specific properties which interact with open sets in different ways, and we can transfer this behavior to the arrows. A Sheaf is how we can think of “collections of functoins” in terms of the arrow-topology; it’s just a thing that “looks like” collections of functions when translated into arrows.
On a topological space, the set of ALL functions on a subset is usually too much. We need to reduce the information in some way in order to do computation. This is where homology and cohomology come in. In topology, differential objects which are NOT the derivatives of already existing functions have extreme importance. So we categorize differential objects which are not derivatives in order to understand the space. This is de Rham Cohomology.
We can mimic this procedure in our arrow-space and with more general sheaves. It requires a lot of effort to make work, but the result is etale cohomology. It allows you to use geometric tools to solve non-geometric problems. Most famously, it was created in order to solve the Weil Conjectures which needed to use the Lefchetz Fixed Point Theorem on objects for which the geometric tools for the Fixed Point Theorem did not immediately apply.
This is as basic as it can be explained. In general, the things which we apply etale cohomology to are arithmetic in nature, and so contain information about prime numbers AS geometry. This can produce problems that we don’t see in normal geometry, so there is still a lot of work going on to figure out how to actually do geometry in a meaningful way in these “prime geometries”. This is called p-adic geometry and is very active, with Peter Scholze winning the Fields Medal a few years ago for the tools he has contributed to the field.
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