The entire species of polar bears descended from one single brown bear that lived in Ireland 20,000 to 50,000 years ago (roughly the peak of the earth's last ice age), according to a study published in Current Biology.

The scientists in this study traced the mitochondrial DNA (which mothers pass unchanged to their offspring) of 242 brown bear and polar bear lineages through the past 120,000 years and came to this Irish 'Eve' conclusion.

They theorize that the cold climate allowed polar bears and various species of brown bears to move into each other's territory, resulting in interbreeding and leading to hybridization.  In the particular case of the brown bear 'Eve' of all modern polar bears, the cold weather possibly allowed a male polar bear to wander all the way to Ireland and breed with her.

Despite the clear scientific evidence of the Irish 'Eve,' the conclusion still surprising given the physical differences between the white swimming polar bears and the brown climbing brown bears.

The explanation is the process of natural selection, whereby the multiple offspring of the 'Eve' brown bear and polar bear hybrid inherited a mixture of characteristics.  Those with polar-bear like characteristics would be favored by their environment and have more offspring.  Eventually, the process produces a breed that's heavily adapted to the Arctic Circle with polar-bear like characteristics.

But how come the entire population of a whole species is descended from one 'Eve,' given that plenty of other polar bears mated at that time? 

This is actually a well-documented phenomenon called the Galton-Watson process.  It explains, for example, why countries with a long history of surnames (like China) only end up with a few surviving ones.