When $\ell \neq p,$ these are rather straightforward to classify (except when $p = 2$);
see Tate's article in the second volume of Corvallis, for example.

The idea is that if $\rho$ is irred., then (unless $p = 2$), it must be induced from a character of a quadratic extension; thus the classification is given by local class field theory for quadratic extensions of $\mathbb Q_p$. (When $p = 2$, there are some exceptional
irreps. that are not induced.)

If $\rho$ is reducible, it is an extension of characters. The characters of $\mathbb Q_p^{\times}$ are classified by local class field theory of $\mathbb Q_p$. There are lots
of ways to compute the possible extensions; Tate local duality/local Euler char. formula gives
one way.

When $\ell = p$, these are classified in terms of etale $(\phi,\Gamma)$-modules. To learn about this, you can e.g. read one of many expository articles on Laurent Berger's website.
(In fact there are many recent papers by Berger, Breuil, and Colmez involving $(\phi,\Gamma)$-modules, all online, and most of them include an introductory page or two recalling the basics of the theory.)

Pete is correct that this $\ell = p$ case is also the starting point of $p$-adic Langlands, just as the case $\ell \neq p$ is related to classical local Langlands.
However, as the above discussion shows, you don't need any Langlands theory to classify these reps.

Added: As JT points out in another answer, the (potentially) semi-stable representations also admit a nice classification, in terms of weakly admissible filtered $(\phi,N)$-modules.

Note that $(\phi,\Gamma)$-modules are themselves pretty nice objects. What is perhaps the most complicated part of the story is how, in the case of a potentially semi-stable representation, one compares its $(\phi,\Gamma)$-module description to its weakly admissible filtered $(\phi,N)$-module description. In the case of crystalline reps., this comparison is made via the theory of Wach modules. In general, it plays an important role in $p$-adic local Langlands, as well as in local Iwasawa theory. Laurent Berger has a number of papers discssing it (beginning with his thesis), and in the case of two-dimensional pst representations it is the subject of the most technical part (Chapter VI) of Colmez's recent long text on $p$-adic local Langlands.