Vesica piscis refers to a specific shape, a 'double-pointed teardrop'. This is sometimes referred to as a mandorla, although technically a mandorla is an almond shape, symmetrical on one axis, while the vesica piscis is symmetrical on two axes. The OED defines it as a "pointed oval shape". It is, essentially, the shape formed by two intersecting circles, the center of a Venn diagram. In fact, it has a very specific definition along these lines:
"The vesica piscis is a type of lens, a mathematical shape formed by the intersection of two disks with the same radius, intersecting in such a way that the center of each disk lies on the perimeter of the other.
This form is popular in art and architecture, particularly in Christian architecture and iconography, and is still popular today as it forms the body of the ichthys. The form was very popular and heavily elaborated on in Gothic architecture, often distorted and incorporated into elaborate mandalas to the point of becoming nearly unrecognizable (and, no doubt, leading to the confusion with mandorlas).
The term 'vesica piscis' literally translates as 'fish bladder' or 'swim bladder' in Latin. The reason for this is debated, but the most common theory is that it is simply named as such because swim bladders are often kind-of-but-not-really-but-sort-of this shape. I personally think this is silly, and agree with John Sidney Hawkins (1758-1842) that it is more likely that this refers to an inflated bladder that has taken on a fish shape. While this sounds odd to modern ears, it is worth noting that the phrase entered the English language, from no certain origin, in 1809; at this time, a common connection would be the pig's bladders used to make rugby balls (elongated ellipsoidal balls, although not as pointed as American footballs). Lacking a better term, these bladders could well be referred to as fish-shaped. The nouveau-Latin was likely added because when historians wished to point out that cathedrals were often designed with the proportions of a vesica piscis (often shortened to piscis), it simply sounded better in Latin.