Proving a theorem, what is meant by sufficiency and necessity?

It is essentially a biconditional, also known as an if and only if.

An "if and only if" statement goes both ways. That is, $p\iff q$ means "if $p$ is true then $q$ is true" and "if $q$ is true then $p$ is true."

The statement "$p$ is sufficient for $q$" means "if $p$ is true, then $q$ is true."

The statement "$p$ is necessary for $q$" means that if we don't have $p$, then we don't have $q$. Therefore, if we have $q$, we certainly have $p$. In other words, "$q$ implies $p$."

When we put the two together, a necessary and sufficient condition is the same as an if and only if.

Consider two statements $A$ and $B$; and we want to know conditions on $A$ for $B$ to be true

Sufficient condition: $A$ is true implies $B$ is true

Necessary conditions: For $B$ to be true, $A$ must be true. It can happen that $A$ is true but $B$ might not be true ( so condition on $A$ is not sufficient).

A condition A is called sufficient for a statement B to hold if A implies B.

A condition A is called necessary for a statement B to hold if B implies A.

"Necessary and sufficient" is the same as equivalent