Why were plane partitions invented? I realize that these objects were originally created by Major Percy Macmahon and today have many applications but what was the original motivation for studying them?
 A: MacMahon invented a technique which he called partition analysis to determine (multivariable) generating functions for many combinatorial objects and as a computational method for solving combinatorial problems in connection with systems of linear diophantine inequalities and equations. This was introduced in his book "Combinatory analysis". At the center was his $\Omega_{\geq}$ operator, for which he proved many properties. He then claimed that plane partitions were a simple toy case to apply these lemmas and was able to compute many interesting generating functions in some limited cases but ran into some problems with the general case of unrestricted plane partitions. He was however lead to some conjectures, some of which he proved later. From there it became clear that there was a lot of interesting mathematics related to plane partitions. I believe you will find some interesting material in the series of papers "MacMahon's partition analysis" I-XII by G.E. Andrews, P.Paule, A Riese and V. Strehl.
Edit: I was a bit rushed to conclude that $\Omega$ had something to do with the motivation to look at plane partitions, see Richard Stanley's answer. I still believe that it was part of the machinery that he built for the same kind of problems that inspired looking at plane partitions. (I mean all of the results about counting tuples of integers satisfying sets of equalities/inequalities.)
A: Gjergji's answer, while it's a correct answer and a good answer, and the question itself only address the very first mention of plane partitions in the literature, and not the almost-first mention that was clearly in the air at the time.  Yes, MacMahon had in mind generating function theory.  However, plane partitions are also a special case of semistandard tableaux, which are used to enumerate basis of irreducible representations of the general linear group.  They are a variation of standard tableau, which enumerate bases of irreps of the symmetric group and were first described by Alfred Young a little bit before MacMahon's work.
I would say that semistandard Young tableaux are more like a fellow traveler of plane partitions than an "application".  MacMahon must have known something about it.  (But conceivably some of these connections were only cleaned up decades later.)
A: It does not seem from MacMahon's first mention of plane partitions that the $\Omega_\leq$ operator was relevant. At the end of Article 42 of his paper "Memoirs on the theory of the partitions of numbers---Part I", MacMahon says "This partition may be termed 'graphically regularised' by reason of its origination in a subjacent succession of lines in the bipartite graph. This species of regularisation is the natural extension to three dimensions of Sylvester's graphical method in two dimensions." He then goes on to develop some simple properties of plane partitions (without using that terminology) and to conjecture his famous generating function $\prod_{n\geq 1}(1-x^n)^{-n}$. He also suggests less confidently that three-dimension partitions have the generating function $\prod_{n\geq 1}(1-x^n)^{-{n+1\choose 2}}$ (now known to be false). The $\Omega_\leq$ operator is used implicitly to prove some simple results, but it does not seem to be relevant to MacMahon's original motivation. I believe that MacMahon did not explicitly use his $\Omega_\leq$ operator until "Memoirs on the theory of the partitions of numbers---Part II", about three years after Part I. In Part II he does consider plane partitions as an example.
A: A quote from "Plane partitions in the work of Richard Stanley and his school" by Christian Krattenthaler addresses this question:

Why were plane partitions so fascinating for MacMahon, and for legions of followers? From his writings, it is clear that MacMahon did not have any external motivation to consider these objects, nor did he have any second thoughts. For him it was obvious that these plane partitions are very natural, as two-dimensional analogues of (linear) partitions (for which at the time already a well established theory was available), and as such of intrinsic interest. Moreover, this intuition was “confirmed” by the extremely elegant product formula in Theorem 1 below. He himself — conjecturally — found another intriguing product formula for so-called “symmetric” plane partitions contained in a given box (see (6.2)).

