A function from $A$ to $B$ assigns exactly one element of $B$ to each element of $A$. The picture below is the whole story: the left set is the domain $A$, the right set is the codomain $B$, and each arrow is the rule.

When is it even a function?

A mapping fails to be a function as soon as one input points to two outputs. In the first panel, one element of $A$ has two arrows leaving it, so it is not a function. Every other panel sends each $A$ to a single $B$, so all of them are functions.

Injective (one-to-one)

A function is injective when no two inputs share an output:

Each element of $B$ gets at most one arrow. Some elements of $B$ can stay untouched, and that is fine. Because no output is reused, you can trace any arrow backwards, which is what lets an inverse exist.

Surjective (onto)

A function is surjective when every element of $B$ gets hit. The image covers the whole codomain:

Several inputs can share the same output; nothing in $B$ is left without an arrow.

Bijective

A function is bijective when it is injective and surjective at the same time: each $A$ pairs with one $B$, and each $B$ with one $A$. This perfect one-to-one pairing is exactly what a true inverse $f^{-1} : B \to A$ needs to exist.