Take a point $x \in X$ and set $y = f(x) \in Y$. Let $X_y = f^{-1}(y)$ the "scheme-theoretic" fiber, i.e. the complex-analytic subspace of $X$ which is cut out the equation $f(x) = y$. Your question is local in $X$, so choose complex coordinates $x_1, \dotsc, x_n$ around $x$ and $y_1, \dotsc, y_m$ around $y$. Then both assertions

1. $x$ a regular point of $f$,
2. $X_y$ is smooth at $x$,

are equivalent to

3. the differential matrix $Df(x) = \left(\frac{\partial f_i}{\partial x_j}\right)_{ij}$ has full rank.

So yes, a point $x$ is critical if and only if the fiber is singular at $x$.

Take a point $x \in X$ and set $y = f(x) \in Y$. Let $X_y = f^{-1}(y)$ the "scheme-theoretic" fiber, i.e. the complex-analytic subspace of $X$ which is cut out the equation $f(x) = y$. Your question is local in $X$, so choose complex coordinates $x_1, \dotsc, x_n$ around $x$ and $y_1, \dotsc, y_m$ around $y$. Then both assertions

1. $x$ a regular point of $f$,
2. $X_y$ is smooth at $x$,

are equivalent to

3. the differential matrix $Df(x) = \left(\frac{\partial f_i}{\partial x_j}(x)\right)_{ij}$ has full rank.

So yes, a point $x$ is critical if and only if the fiber is singular at $x$.