For example, when we use the function notation $f: \R \to \R$, we mean that $f$ is a function from the real numbers to the real numbers. In other words, the codomain of $f$ is the set of real numbers $\R$ (and its set of possible inputs or domain is also the set of real numbers $\R$).
Just because an object is in the codomain of a function, it does not necessarily mean that there is an input for which the function will output that object. For example, we could define a function $f: \R \to \R$ as $f(x)=x^2$. Since $f(x)$ will always be non-negative, the number $-3$ is in the codomain of $f$, but there is no input of $x$ for which $f(x)=-3$. The set of all outputs that result from putting all inputs into the function is called the range. For the above $f$, the range is the set of non-negative real numbers while the codomain is the set of all real numbers.