What is the operation $\boxtimes$? Reading papers about $p$-adic analysis and Galois representations, I have found objects like this $D \boxtimes \mathbb{Q}_p$. So my question is what is $\boxtimes$ and how do we read it ?
 A: In the context of Colmez's papers, the notation has its own meaning, not related (by more than vague analogy) to other meanings it has in other contexts where it is used.  
You will have to read Colmez's article in Asterisque 330 to learn the details.
Roughly: you should think of the $(\varphi,\Gamma)$-module as being an object (like a space of measures, or functions) living over $\mathbb Z_p$.  Then
$D\boxtimes \mathbb Q_p$ is what you get by using scaling by $p$ (which is rigorously defined using the operator $\psi$) to "stretch" the $(\varphi,\Gamma)$-module out over $\mathbb Q_p$. 
Similarly $D\boxtimes \mathbb P^1$ is what you by taking two copies of $D$ and gluing them together, in accordance with the way that $\mathbb P^1(\mathbb Q_p)$ is obtained by gluing together two copies of $\mathbb Z_p$.

Non-mathematical remark:
I should add that what you are asking about is very recent mathematics, and has a pretty high entry-level.  Where/with who are you learning this material?  You may be better off asking your advisor directly rather than trying to learn this on math.SE. 
You may also want to look at some of Colmez's lectures, several of which should be available online.  He lectured this past July at the Durham conference, and I believe those lectures were videotaped.  In the past he has lectured at Luminy (several times, I think), at the Newton Institute (Summer of 09, if I remember correctly), and this past March he gave a lecture course at the IAS (although I wasn't there, so I don't know if it was filmed).
You may also find it easier to study the functor from $GL_2$-reps. to Galois reps. before trying to go backwards from Galois reps. to $GL_2$-reps. (which is the point of the $\boxtimes$ constructions).  As well as Colmez's Asterisque 330 article, there is also my short preprint On a class of coherent rings ..., which you will be able to find with a google search.
A: There is a notion of external (also called exterior, or box) tensor product $\boxtimes$ ( e.g. http://books.google.com/books?id=6GUH8ARxhp8C&pg=PA24 ).  
I think that the usage is not completely standardized, in that the definition is often adapted to other contexts (examples at https://mathoverflow.net/search?q=boxtimes ), but the adaptations are not always consistent with each other.
A: This is probably not directly what you're asking about, but it might be related.  In any event it can't hurt to add it.
Let $X$ be a topological space and let $E_1 \to X$, $E_2 \to X$ be vector bundles (or sheaves, probably).  One often defines $S_1 \boxtimes S_2$ to be the bundle $\pi_1^\ast E_1 \otimes \pi_2^\ast E_2^\ast$ over $X \times X$, where $\pi_1, \pi_2: X \times X \to X$ are the usual projection maps.  Thus a vector over the point $(p,q)$ is a linear map from $S_1(p)$ to $S_1(q)$.  This bundle is useful in differential geometry because its sections are Schwarz kernels of linear operators.  I have seen it arise in algebraic geometry (over $\mathbb{C}$) as well for a similar reason.  Perhaps there is an analogy between the notation coming from geometry and the notation coming from representation theory and number theory?
