= Solution
A point $y_0\in Y$ is a <regular value> of the smooth map $f:X\to Y$ if, for every $x\in f^{-1}(y_0)$, the derivative
$$
D f_x:T_xX\longrightarrow T_{y_0}Y
$$
is surjective. A value outside the image is regular vacuously.
For a nonvacuous example, take
$$
f:\mathbb R^2\to\mathbb R,
\qquad f(x,y)=x^2+y^2.
$$
The value $1$ is regular: its fibre is the unit circle, and at every $(x,y)$ on that circle,
$$
D f_{(x,y)}(u,v)=2xu+2yv
$$
is a nonzero linear functional and hence surjective.
<Sard theorem> states that the set of critical values of a smooth map between smooth manifolds has measure zero in the target. Equivalently, almost every target point is a regular value.
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