I was reading Stefano's answer and the discussion that followed (in the comments) and I thought I'd sum this up below. Hope somebody finds it useful some time!
The affine cone over a projective variety $X \subset \mathbb{P}^n$ is the set $$C(X):=\{0\}\cup p^{-1}(X)\subset \mathbb{C}^{n+1},$$ where $p^{-1}(X)\subset \mathbb{C}^{n+1}\setminus\{0\}$ is the pre-image of $X$ under the canonical projection $$p: \mathbb{C}^{n+1}\setminus\{0\} \longrightarrow \mathbb{P}^n$$ sending a non-zero vector $(x_0,x_1\,\ldots\,,x_n)$ to the line it generates (this line being usually denoted by $[x_0:x_1:\,\ldots\,:x_n]$ in homogeneous coordinates).
This set $C(X)$ is indeed a cone in the vector space $\mathbb{C}^{n+1}$, in the sense that it is stable by multiplication under a non-zero complex number: if $x:=(x_0,x_1\,\ldots\,,x_n)\in p^{-1}(X)$, then $$\forall\, \lambda \in \mathbb{C}^*, \lambda x = (\lambda x_0, \lambda x_1\,\ldots\,,\lambda x_n)\in p^{-1}(X),$$ as follows from the definition of $p$.
Note that if $X\subset \mathbb{P}^n$ has dimension $d$, then its affine cone $C(X)\subset \mathbb{C}^{n+1}$ will have dimension $d+1$.
For instance, the homogeneous equation $x-y+z=0$ defines a line $L$ in $\mathbb{P}^2$, whose affine cone $C(L)$ is the plane of equation $x-y+z=0$ in $\mathbb{C}^3$.
To see that the homogeneous equation $x-y+z=0$ defines a line in $\mathbb{P}^2$, you can look at it in an affine chart. The three standard affine charts on $\mathbb{P}^2$ are defined on the open sets $U_x=\{x\neq 0\}$, $U_y=\{y\neq 0\}$ and $U_z=\{z\neq 0\}$. Explicitly, these are:
\begin{align}
\phi_x([x:y:z]) = \left(\frac{y}{x},\frac{z}{x}\right),\\
\phi_y([x:y:z]) = \left(\frac{x}{y},\frac{z}{y}\right),\\
\phi_z([x:y:z]) = \left(\frac{x}{z},\frac{y}{z}\right).
\end{align}
The respective images of $L=\{[x:y:z]\in\mathbb{P}^2\ |\ x-y+z=0\}$ under each one of these charts are:
\begin{align}
\phi_x(L) = \{(y,z)\in\mathbb{C}^2\ |\ 1-y+z=0\},\\
\phi_y(L) = \{(x,z)\in\mathbb{C}^2\ |\ x-1+z=0\},\\
\phi_z(L) = \{(x,y)\in\mathbb{C}^2\ |\ x-y+1=0\}.
\end{align}
The composition of the projection $p$ and the chart $\phi_x$ induces a bijection between the line $C(L)\cap \{x=1\}$ in $\mathbb{C}^3$ (intersection of two planes) and the line $\phi_x(L)=\phi_x \circ p((C(L))$ in $\mathbb{C}^2$. And similarly for $\phi_y$, $\phi_z$ and the planes $\{y=1\}$, $\{z=1\}$ in $\mathbb{C}^3$.
By the projective Nullstellensatz (over $\mathbb{C}$ or over an algebraically closed field), a projective variety $X\subset\mathbb{P}^n$ is defined by a system of homogeneous equations $$\{f_1=0,\,\ldots\,,f_m=0\}$$ (meaning that each $f_i:\mathbb{C}^{n+1}\longrightarrow \mathbb{C}$ is a homogeneous polynomial). And then the affine cone $C(X)$, as a subset of $\mathbb{C}^{n+1}$, is defined by the same equations. In particular, it is an affine variety.
In Stefano's example, $$X=\{[x:y:z]\in\mathbb{P}^2\ |\ xy-z^2=0\}.$$ This is a curve in $\mathbb{P}^2$, whose images in the standard affine charts are the curves:
\begin{align}
\phi_x(X) = \{(y,z)\in\mathbb{C}^2\ |\ y-z^2=0\},\\
\phi_y(X) = \{(x,z)\in\mathbb{C}^2\ |\ x-z^2=0\},\\
\phi_z(X) = \{(x,y)\in\mathbb{C}^2\ |\ xy-1=0\}.
\end{align}
These curves are plane quadrics (degree $2$ equation in two variables), also known as conics.
The affine cone of $X$ is $$C(X) = \{(x,y,z)\in\mathbb{C}^3\ |\ xy-z^2=0\},$$ which is the equation of a (quadric) surface in $\mathbb{C}^3$ (degree $2$ equation in three variables). And the intersections of this surface with the lines $\{x=1\}$, $\{y=1\}$ and $\{z=1\}$ in $\mathbb{C}^3$ are curves that are in bijection respectively with $\phi_x(X)$, $\phi_y(X)$ and $\phi_x(X)$.
