I'm confident there is a better, albeit more technical, explanation for this condition. That is how she explained it to me. Don't know how common is this condition.
That's called a branchial cyst. It has nothing to do with gills. While we are developing in the womb, for a while we have several sets of structures called pharyngeal arches. Bunches of structures in the neck and face develop from the pharyngeal arches. And it's way more complicated than I could do justice to it to explain (or than you want to hear), but let's just say that if all does not happen as it's supposed to happen, there can wind up being a cyst in the front of the side of the neck. Branchial cysts are pretty harmless, and they're usually only discovered in adulthood, because they can enlarge.
It's called a branchial cleft cyst. The docs use the word "gill" because it's descriptive to lay people since the clefts resemble gills in the embryo. As it turns out, both fish gills AND human parathyroid glands develop from these clefts.
"Branchial pouches" is synonymous with "pharyngeal pouches" which is the term used in this article. "Branchial" = "Of, relating to, or resembling the gills of a fish, their homologous embryonic structures, or the derivatives of their homologous parts in higher animals breathing" (AHED).