Suppose I have a finite pointed CW complex X and an integer $n$. In the context of stable homotopy theory this is known as a finite spectrum. For a given prime $p$, something called the $p$-completion $X_p$ of $(X,n)$ is referred to in the following paper. ("Let $X_p$ be the $p$-completion of a spectrum $X$" at the top of the second page.)
How is this defined exactly? Does it have a well-established meaning in topology? I cannot find anything that explains this adequately.
This is explained thoroughly in Chapter 9 of Spectra and the Steenrod Algebra by Margolis. In particular, a $p$-completion of a spectrum $X$ is defined to be a map $f: X \to Y$ such that $\pi_* f: \pi_* X \to \pi_* Y$ gives the $p$-completion of the homotopy groups of $X$. Margolis notes that this need not exist in general, but it does if you assume that for each $n$, $\pi_n X$ is a finitely generated abelian group. Call such an $X$ "finite type". (This is Theorem 1 in Chapter 9.) Combining Proposition 9 and Theorem 11 gives: if $X$ is the $p$-localization of a finite-type spectrum, then its $p$-completion may be obtained as Bousfield localization with respect to the Moore spectrum $S(\mathbb{Z}/p)$ (as @user10354138 said).