 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.