The beautiful analogy between number fields and function fields (and in general, algebra and geometry) that one learns about in arithmetic algebraic geometry.
Specifically, the
Some specific examples:
The idea that a Galois group and a fundamental group (one algebro-number theoretic, the other geometric/topological) are two instances of the same thing.
The use of the term ramification in both number theory and geometry. Describing $\mathbb{Z}$ as simply connected because $\mathbb{Q}$ has no unramified extensions.
The appearance of integral closure in both algebraic geometry and algebraic number theory. The integral closure, in the former case, actually corresponds to a distinct geometric idea: non-singularity.
The idea of considering a prime number to be a point; then viewing localization at that prime, -adic completion at that prime, and the residue field of that prime as if they were the corresponding geometric objects. In particular, using the term "local" in number theory, as if we were talking about geometry!
There's much more, and probably a lot more that can be said if one knows more advanced algebraic geometry, especially This idea is built into scheme theory.
There are many more examples.
This book looks deeply into the relationships between Galois groups and fundamental groups and eventually develops a theory which covers both.
This book explores the beautiful relation between algebraic curves and algebraic number theory.
