For many years, pale skin was associated with wealth and beauty (in European cultures, not sure about others) and women in particular went to extreme lengths to make themselves look pale like using white lead-containing makeup. At what point and why was this fashion which lasted centuries changed so we now desire tanned skin?
It is generally agreed that Coco Chanel popularized tanning after she was accidentally sunburnt in French Rivera.
To expand on that, swimming for women and going to the beach for vacation only became popular in the second half of the 1800's due to the rise of the middle class. In the beginning women wore long sleeved bathing suit with long stockings to avoid tanning (like what Mrs. Lovette wore in the movie Sweeney Todd). Gradually long stockings and long sleeves were abandoned but bathing suit was still rather bulky. By the early 20's tightly fitted one piece bathing suit became popular (but many cities banned this kind of bathing suit). With bathing suits exposing more and more skin and more and more fashionable women going to the beaches tanning became unavoidable. Instead of trying to avoid tanning in vain, fashionable women decided to adapt tanning instead to signify that they go to the beach.
Source: This Fabulous Century, 1890, 1900-1920, and 1920-1930