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 

$\textrm{crk}(f):=n-\textrm{rank}(Hessian)(f)(o)$ 

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.