For any commutative ring $A$ with unit $1_A$, we have a ring homomorphism $\varphi\colon\mathbf Z\to A$ de fined by $\varphi(m)=m\cdot1_A$. The kernel of this homomorphism is an ideal of $\mathbf Z$, generated by a unique element $n\in\mathbf N$, and $\varphi$ factors as
$$\mathbf Z\longrightarrow\mathbf Z/n\mathbf Z\hookrightarrow A.$$
Now, if $A$ is an integral domain, the ideal $n\mathbf Z$ is a prime ideal, so $n$ is either $0$ or a prime number, called the characteristic of the domain. This is the case in particular if $A$ is a field $F$.
If $F$ has characteristic $0$, $\varphi$ extends to a field homomorphism from the field of fractions $\mathbf Q$ of $\mathbf Z$ to $F$, so that $F$ (and all its subfields) contains a subfield isomorphic to $\mathbf Q$.
If $F$ has characteristic $p>0$, $\varphi$ induces a field homomorphism from the field $\mathbf F_p=\mathbf Z/p\mathbf Z$ to $F$, so $F$ and all its subfields contain a subfield isomorphic to $\mathbf F_p$.
In all cases this smallest subfield of $F$ is called its prime subfield.