= Solution
A <regular function> on $U$ is a function $f:U\to k$ locally representable, in an <affine chart> of the ambient <projective space>, as a quotient $a/b$ of <polynomial functions> with denominator nonzero on that <neighbourhood>. Equivalently it is locally a quotient of homogeneous forms of the same degree. The equal-degree condition makes the quotient independent of the chosen projective representative.
The <ring> $\mathcal O_{X,p}$ is the <ring> of <germs> of such functions near $p$. Two representatives $(U,f)$ and $(V,g)$ define the same <germ> when they agree on some smaller <neighbourhood> of $p$. Addition and multiplication use a common <neighbourhood>. It is a <local ring>: its unique <maximal ideal> consists of <germs> vanishing at $p$, and any <germ> nonzero at $p$ has a regular reciprocal on a smaller <neighbourhood>.
$$
\boxed{\mathfrak m_p=\{[f]_p:f(p)=0\},\qquad\mathcal O_{X,p}/\mathfrak m_p\cong k.}
$$
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