On the algebra of functions of an embedded manifold

Question

We know that we can embed a manifold $\mathcal{M}$ of dimension $n$ in $\mathbb{R}^m$ with $m$ sufficiently high and specify the embedding using $n-m$ relations for the ambient coordinates. The algebra of functions on $\mathcal{M}$, $C(\mathcal{M})$, is the quotient of the algebra of functions on $\mathbb{R}^m$ by the ideal generated by the $n-m$ relations. What are concretely this ideal and the equivalence relation that gives rise to the quotient? Can we understand this, for example, for $S^2$ embedded in $\mathbb{R}^3$ via $x^2+y^2+z^2=1$? Thanks.