the regular-value theorem
How do you get your hands on a manifold? Often the easiest way is to cut one out as a solution set: the points where some smooth equations hold. The regular-value theorem is the licence that says this works — if the equations are 'non-degenerate' at every solution point, the solution set is automatically a smooth submanifold, with its dimension and tangent spaces handed to you for free.
Let f: M -> N be smooth and let c in N be a regular value, meaning df_p is surjective at every p in the level set f^{-1}(c) (vacuously regular if the level set is empty). Then f^{-1}(c) is an embedded submanifold of M of dimension dim M minus dim N, and its tangent space at each p is the kernel of df_p. The reason is the constant-rank theorem: surjectivity of df at the level set makes f a submersion there, so locally f looks like a projection and its fibers look like coordinate slices. In the common case N = R^k, c is regular iff the gradients of the k component functions are linearly independent at every common zero.
This is the workhorse that constructs most explicit manifolds. The unit sphere S^{n-1} is f^{-1}(1) for f(x) = |x|^2, whose differential 2x is surjective off the origin, and 1 is a regular value, so S^{n-1} is an (n-1)-manifold. The orthogonal group O(n) is the level set of the map A -> A^T A at the identity matrix, which one checks is a regular value, giving O(n) its manifold structure. The essential caveat is the word regular: at a critical value the theorem says nothing, and the level set can be singular — for f(x, y) = x^2 - y^2 the level 0 is two crossing lines, not a manifold, because 0 is a critical value where the gradient vanishes at the origin.
For f(x, y, z) = x^2 + y^2 + z^2 on R^3, the differential (2x, 2y, 2z) is surjective everywhere except the origin, so every c > 0 is a regular value and f^{-1}(c) is a smooth 2-sphere of radius sqrt(c). The value c = 0 is critical and its level set is a single point, not a surface.
Every positive value is regular for |x|^2, carving out a sphere.
Regularity is required only on the level set f^{-1}(c), not on all of M. And a critical value may still have a manifold preimage by accident — regularity is sufficient, not necessary, for the level set to be a manifold.