There have been many definitions of the "geoid" over 150 years or so. Here is the one currently adopted at NGS:
geoid: The equipotential surface of the Earth's gravity field which best fits, in a least squares sense, global mean sea level
Even though we adopt a definition, that does not mean we are perfect in the realization of that definition. For example, altimetry is often used to define "mean sea level" in the oceans, but altimetry is not global (missing the near polar regions). As such, the fit between "global" mean sea level and the geoid is not entirely confirmable. Also, there may be non-periodic changes in sea level (like a persistent rise in sea level, for example). If so, then "mean sea level" changes in time, and therefore the geoid should also change in time. These are just a few examples of the difficulty in defining "the geoid".
1828: C.F. Gauss first describes the "mathematical figure of the Earth" (Heiskanen and Moritz, 1967, p. 49 ; Torge, 1991, p. 2 ; Gauss, 1828)
1849: G.G. Stokes derives the formula for computing the "surface of the Earth's original fluidity" from surface gravity measurements. This later became immortalized as "Stokes's integral" (Heiskanen and Moritz, 1967, p. 94; Stokes, 1849)
1873: J.F. Listing coins the term "geoid" to describe this mathematical surface (Torge, 1991, p. 2 ; Listing, 1873)
1880: F.R. Helmert presents the first full treatise on "Physical geodesy", including the problem of computing the shape of the geoid.
Gauss, C.F., 1828: Bestimmung des Breitenunterscchiedes zwischen den Sternwarten von Gottingen und Altona, Gottingen.
Heiskanen, W.A. and H. Moritz, 1967: Physical Geodesy, W.H. Freeman, San Francisco, 364 pp.
Helmert, F.R., 1880: Die mathematischen und physicalischen Theorien der hoheren Geodasie, Teubner, Leipzip, Frankfurt.
Listing, J.B., 1873: Uber unsere jetzige Kenntnis der Gestalt und Grosse der Erde, Nachr. d. Kgl., Gesellsch. d. Wiss. und der Georg-August-Univ., 33-98, Gottingen.
Stokes, G.G., 1849: On the variation of gravity at the surface of the Earth, Transactions of the Cambridge Philosophical Society, V. 8, p. 672.
Torge, W., 1991: Geodesy, Walter de Gruyter, Berlin, 264 pp