To expand a little on Allen's useful answer:

1) The question doesn't specify over which field (presumably $\mathbb{R}$ or $\mathbb{C}$) the f.d. irreps of $G=GL_n(\mathbb{R})$ should be studied, but in this particular case it doesn't matter much. From the viewpoint of Steinberg's 1967-68 Yale lectures, the *special* linear group is a *Chevalley group*; so its structure and representations in characteristic 0 can be studied first over $\mathbb{Q}$ followed by extension of scalars.

2) The f.d. representation theory of semisimple, or more generally reductive, Lie groups has been well understood for a century, going back to work of Elie Cartan and Hermann Weyl. But most of the difficulty involves *simple* groups (semisimple with finite center), while passage to general linear groups over $\mathbb{C}$ just involves easy modifications by powers of the determinant. So the textbook literature has mostly concentrated on simple Lie groups, though books for physicists usually focus more on $GL_n(\mathbb{C})$ and the combinatorics of partitions along with Schur-Weyl duality. Adapting this to the real group $G$ is fairly straightforward, as Allen indicates.

3) Anyway, it's not easy to specify a standard source for the specific question raised here. That's partly because modern research efforts have focused mostly on the less-understood world of infinite dimensional representations.

4) The question asks for a "list" and a "construction". Allen basically gives a list, taking for granted the classical parametrization of f.d. irreps of the simple groups over $\mathbb{C}$ (or restrictions to $\mathbb{R}$) by highest weights in general or partitions in this special case. But constructions are not so easy even for general or special linear groups, where the best concrete approach seems to be via Schur-Weyl duality.

`$(\mathbb{g}, K)$`

-modules were developed for real Lie groups as a tool in studying infinite dimensional representations; for the finite dimensional case classical methods are enough. – Jim Humphreys Mar 28 '11 at 14:09