A meteoroid penetrating the Earth's atmosphere leaves behind a trail of dense plasma embedded in the lower E/upper D region ionosphere. While radar measurements of meteor trail evolution have been collected and used to infer meteor and atmospheric properties since the 1950s, no accurate quantitative model of trail fields and diffusion exists. This paper describes a two-dimensional (2-D) analytical theory of meteor trail plasma physics that fills the gap between the studies of dense trail physics without background plasma, 1-D models, and those for small disturbances. Major new results include an estimate of the spatial distribution of a trail's ambipolar electric potential and a quantitative model of diffusion rates, both parallel and perpendicular to the geomagnetic field, with a prediction that in the course of their diffusion, dense trails transform from anisotropic to more isotropic. These results are important for interpreting specular and nonspecular radar measurements of meteor trails and for accurate modeling of plasma instabilities. A companion paper shows that the results from this analytical theory agree well with simulations results.