Why study the p-completions of a space? Given a nice topological space $X$ there are various notions of a 'completion' at a set of primes. Some of the most common constructions may be found in Bousfield-Kan's, May's, Neisendorfer's or Sullivan's classic textbooks - that last three of which I have read. 
But what information about a space is contained in its p-completions? Whilst I have a good handle on the information that may be gleamed from study of the p-localizations, the slightly more abstract nature of the completion leaves me unsure of how to interpret its properties. I am aware of the various arithmetic squares and have no trouble with the concepts or algebra, I just lack any concrete examples of the theory of completions yielding any useful information about the homotopy type of a given space.
 A: First one should separate between the property and being $p$-complete and process of $p$-completion. In the classical setting, the $p$-completion functor is not so well-behaved for general spaces. For example, the $p$-completion of a space need not be $p$-complete. One way to remedy this is to notice that $p$-completion is not really a functor that should take values in spaces. To understand why, consider the analogous case of groups. The pro-$p$ completion of a group should really be consider as a pro-finite group, as apposed to an ordinary group. This additional structure can be encoded either via a suitable topology on the group, or by replacing the pro-finite group with its inverse system of finite (continuous) quotients. The latter description turns out to fit in a more general categorical context. The collection of "inverse systems of finite groups" can be organized into a category, which is called the pro-category of the category of finite groups. This is a general categorical construction which associates to a category $C$ the category $Pro(C)$ whose objects are inverse systems of objects in $C$ and whose morphisms are suitably defined. We have a natural fully-faithful embedding $C \longrightarrow Pro(C)$ which exhibits $Pro(C)$ as the free category generated from $C$ under cofiltered limits. Furthermore, if $C$ has finite limits then $Pro(C)$ has all small limits. Now, given categories $C,D$ which have finite limits, and a functor $f:C \longrightarrow D$ which preserves finite limits, we obtain an induced functor $Pro(f):Pro(C) \longrightarrow Pro(D)$ which preserves all limits. Under suitable additional conditions (for example, if $C,D$ and $f$ are accessible), the functor $Pro(f)$ will admit a left adjoint $G: Pro(D) \longrightarrow Pro(C)$. A classical example is when $C$ is the category of finite groups, $D$ is the category of all groups, and $f: C \longrightarrow D$ is the natural inclusion. In this case, the corresponding left adjoint $G: Pro(D) \longrightarrow Pro(C)$, when restricted to $D$, is exactly the pro-finite completion functor. If we replace $C$ with the category of finite $p$-groups then we obtain the pro-$p$-completion functor.
A similar situation occurs with spaces. Recall that a $p$-finite space is a space with finitely many connected components, each of which has finitely many non-trivial homotopy groups, and all the homotopy groups are finite $p$-groups. Let $\mathcal{S}_p$ be the $\infty$-category of $p$-finite spaces, $\mathcal{S}$ the $\infty$-category of spaces and $f: \mathcal{S}_p \longrightarrow \mathcal{S}$ the natural inclusion. The induced left adjoint $G:Pro(\mathcal{S}) \longrightarrow Pro(\mathcal{S}_p)$, when restricted to $\mathcal{S} \subseteq Pro(\mathcal{S})$, is in some sense the more correct version of the $p$-completion functor. In particular, if $X$ is a space, then the $p$-completion should really be considered as an inverse system of $p$-finite spaces, and not a single space. The inverse limit of this system then coincides with the classical $p$-completion. However, for many reasons it is better to consider the inverse system itself. For example, unlike the classical $p$-completion functor, the functor $G:Pro(\mathcal{S}) \longrightarrow Pro(\mathcal{S}_p)$ is a localization functor with respect to $\mathbb{Z}/p$-cohomology (of pro-spaces). As such, the functor $G$ is idempotent, in the sense that $G(G(X)) = G(X)$, a property that is not shared by the classical $p$-completion functor. Furthermore, the answer to the question "what information on $X$ is contained in $G(X)$" has a precise answer now. It is exactly all the information concerning maps from $X$ to $p$-finite spaces.
In light of the enhanced version of the $p$-completion functor, one might ask what does it mean for a space to be $p$-complete. Going back to the situation of groups, one may observe that some groups have the property that they are isomorphic to the underline discrete group of their pro-$p$ completion. In terms of pro-objects, some groups are isomorphic to the inverse limit of their pro-$p$-completion, realized in the category of groups. For example, the group $\mathbb{Z}_p$ of $p$-adic integers has this property. In this case, the group itself is completely determined by its $p$-finite quotients. Similarly, a space is $p$-complete when it is equivalent to the realization of its (enhanced) $p$-completion in the $\infty$-category of spaces. This property has several equivalent manifestations. One of them is the following. For each space $X$, we may consider the cochain complex $C^*(X,\overline{\mathbb{F}}_p)$ with values in the algebraic closure $\overline{\mathbb{F}}_p$ of the finite field $\mathbb{F}_p$. It turns out that $C^*(X,\overline{\mathbb{F}}_p)$ carries a natural structure of an $E_\infty$-algebra over $\overline{\mathbb{F}}_p$. The construction $X \mapsto C^*(X,\overline{\mathbb{F}}_p)$ can then be considered as a functor from spaces to the opposide category of  $E_\infty$-algebras over $\overline{\mathbb{F}}_p$. This functor admits a right adjoint, sending an $E_\infty$-algebra $R$ to the mapping space $Map_{E_\infty-Alg}(R,\overline{\mathbb{F}}_p)$. For every space $X$ we then obtain a unit map $X \longrightarrow Map_{E_\infty-Alg}(C^*(X,\overline{\mathbb{F}}_p),\overline{\mathbb{F}}_p)$. It turns out that $X$ is $p$-complete precisely when this unit map is an equivalence. This means that the functor $X \mapsto C^*(X,\overline{\mathbb{F}}_p)$ is fully-faithful when restricted to $p$-complete spaces and we can hence consider $p$-complete spaces as a suitable full sub-category of the opposite category of $E_\infty$-algebras. In addition to the conceptual importance of this result, it also has practical applications. For example, it means that we may construct an Adams-type spectral sequence to compute the homotopy groups of $X$ by resolving $C^*(X,\overline{\mathbb{F}}_p)$ into free $E_\infty$-algebras.
A: I'll give an answer from the point of view of an algebraic topologist,
somebody who cares about examples and computations.  This all goes way back 
and has nothing to do with modern generalities.  First off, calculationally, algebraic 
topologists can compute, like to compute, and need to compute mod p homology.  It is 
far more accessible than p-local homology, and it is the focus of completion, not localization, 
at p.  
An elementary concrete example of 
homotopy information invisible to p-localization is that $K(\mathbb Z/p^{\infty},n)$ 
becomes equivalent to $K(\mathbb Z_p,n+1)$ after $p$-completion.  This jacks up to Quillen's
equivalence between the $p$-completions of the algebraic $K$-theory space $K(\bar{\mathbb F}_q)$ and $BU$ where $p\neq q$.  The homotopy groups of $K(\bar{\mathbb F}_q)$ are 
$\oplus_{p\neq q} \mathbb Z/p^{\infty}$ in odd degrees and $0$ in even degrees.
The homotopy groups of $BU$ are $0$ in odd degrees and $\mathbb {Z}$ in even degrees.
Obviously $p$-localization knows nothing whatsoever about this equivalence.   
Calculationally,
the classical Adams spectral sequence converges to the stable homotopy groups of
the $p$-completion, not the $p$-localization, of the space (or spectrum) one 
starts with; that is also true unstably. The Atiyah-Segal completion theorem says that
the topological $K$-theory of $BG$ is the completion of $R(G)$ at its augmentation
ideal.  If $G$ is a $p$-group, this is just $p$-adic completion (on the reduced level). 
That is a place where use of pro-groups helps calculation, but in most of the 
calculational applications use of pro-objects is not especially helpful.  
There are tons of places where $p$-completion is essential to the proof or statement 
of results: the Sullivan conjecture, the Segal conjecture, the classification of 
$p$-compact groups, the study of atomic spaces, etc.  It is $p$-completion that is 
relevant to all of these and many more.  
A technical reason $p$-completion is so convenient is that there are no phantom maps 
to worry about if one restricts to nilpotent spaces of finite type.  That is also 
true for rationalization but not for localization at $p$.
A more sophisticated example is Mandell's algebraization of $p$-adic homotopy theory, the closest analogue we have of the classical algebraization of rational homotopy theory.
(The last part of Harpaz's answer concerns this.) 
I could go on for many pages.  The more algebraic topology one learns, 
the more frequently $p$-completion appears and the more natural it seems. 
It is worth adding that we do not understand $p$-completion very well for
non-nilpotent spaces, where there are several very different notions,
none well understood calculationally.  This deserves more study.
