Examples:

1. Let $L=\Rats(\sqrt{2})$ The extension $L/\Rats$ is an algebraic extension. Indeed, any element $\alpha\in L$ is of the form $$\alpha=q+t\sqrt{2}\in L$$ for some $q,t\in\Rats$ Then $\alpha\in L$ is a root of $$X^2-2qX+q^2-2t^2=0$$
2. The field extension $\Reals/ \Rats$ is not an algebraic extension. For example, $\pi\in \Reals$ is a transcendental number over $\Rats$ (see pi). So $\Reals/\Rats$ is a transcendental extension of fields.
3. Let $K$ be a field and denote by $\overline{K}$ the algebraic closure of $K$ Then the extension $\overline{K}/K$ is algebraic.
4. In general, a finite extension of fields is an algebraic extension. However, the converse is not true. The extension $\overline{\Rats}/\Rats$ is far from finite.
5. The extension $\Rats(\pi)/\Rats$ is transcendental because $\pi$ is a transcendental number, i.e. $\pi$ is not the root of any polynomial $p(x)\in \Rats[x]$