Blowup of the affine plane at the origin
= Blowup of the affine plane at the origin
The blowup of $\mathbb A^2$ at the origin is
$$
\operatorname{Bl}_0\mathbb A^2
=\{((X,Y),[W:Z]):XZ=WY\}\subseteq\mathbb A^2\times\mathbb P^1.
$$
Its projection to $\mathbb A^2$ is an isomorphism away from the origin, while the fiber above the origin is the exceptional curve $\mathbb P^1$.
It is an explicit example of <blowup of an algebraic variety>.