Let me stress that I am only interested in $p$-adic fields in this question, for reasons that will become clear later. Let me also stress that in some sense I am basically assuming that the reader knows what the "1970s version of the local Langlands conjectures" are when writing this question---there are plenty of references that will get us this far (I give one below that works in the generality I'm interested in).
So let $F$ be a finite extension of $\mathbf{Q}_p$, let $G$ be a connected reductive group over $F$, let $\widehat{G}$ denote the complex dual group of $G$ (a connected complex Lie group) and let ${}^LG$ denote the $L$-group of $G$, the semi-direct product of the dual group and the Weil group of $F$ (formed using a fixed algebraic closure $\overline{F}$ of $F$).
Here is the "standard", or possibly "standard in the 1970s", way of formulating what local Langlands should say (for more details see Borel's paper "Automorphic $L$-functions", available online (thanks AMS) here at the AMS website. One defines sets $\Phi(G)$ ($\widehat{G}$-conjugacy classes of admissible Weil-Deligne representations from the Weil-Deligne group to the $L$-group [noting that "admissible" includes assertions about images only landing in so-called "relevant parabolics" in the general case and is quite a subtle notion]) and $\Pi(G)$ (isomorphism classes of smooth irreducible admissible representations of $G(F)$), and one conjectures: