I think that students should know about the structure of mathematics, by which I mean how and why definitions and theorems are written and what their parts mean. Most of the examples are from calculus, but the ideas apply to all parts of mathematics.

Theorems and Axioms

Teaching and Learning Theorems  and another Theorems

Definitions and another  Definitions

For Any – For Every – For All

Then there is this – Existence Theorems

Proof

Units How to find the units of a derivative or a definite integral