At least over $\mathbb{C}$, there is a simple answer.
A plane curve $f(x,y)=0$ has a singularity of type $A_k$ in $o=(0,0)$ if and only if
$o$ is a $double$ $point$, that is all first partial derivatives of $f$ vanish in $o$ but there is at least one second partial derivative which is not zero;
the $Milnor$ $number$
$\mu(f, o):= \dim_{\mathbb{C}}\mathcal{O}_{o}/(f_x, f_y)$
is equal to $k$. Here $\mathcal{O}_{o}$ denotes the ring of convergent power series.
This can be generalized in higher dimensions. In fact, one proves that a (germ of) complex hypersurface singularity $f(x_1, ...,x_n)=0$ is of type $A_k$ if and only if
- the corank
$\textrm{crk}(f):=n-\textrm{rank}(\textrm{Hessian}(f))(o)$
is $ \leq 1$;
- the Milnor number
$\mu(f, o):= \dim_{\mathbb{C}}\mathcal{O}_{o}/(J_f)$
is equal to $k$.
This follows from a sort of generalized Morse Lemma. See the book GREUEL - LOSSEN - SHUSTIN "Introduction to singularities and deformations" p. 150 for the proof.