Arrows have been used to communicate an enormous variety of different concepts. They can represent almost any relation between two objects (or sets of objects). For clarity, when different relations must be shown in close proximity to one another, they are depicted by visually distinct arrows; over time many specific styles of arrow have acquired particular meanings.
These meanings are often idiomatic to a particular field: the curly arrows that indicate electron movement, or the double arrows showing that a reaction stops at equilibrium rather than going to completion, are not used outside of chemistry. The US military uses a large set of arrows (at least 30) to depict specific troop movements and attacks in their tactical mission graphics. Chemists use fewer: I have tabulated some of the most commonly used below.
Some arrows have very different meanings in different contexts: in mathematics, the double arrow represents logical implication (‘if X is true, then Y must be true too’); in chemistry, it is the retrosynthetic arrow, (‘X can be produced from Y’).
A few arrows do not represent relationships, but things themselves. For example, an arrow may represent a vector (such as a force or velocity) in a diagram.