Like weight gain, age progression is taboo because it involves attaching an erotic element to something our society is hell-bent on avoiding. This taboo is part of the allure in many stories.
Age progression typically involves an acceleration of the aging process, by magical, scientific, or unexplained methods. This results in graying hair, wrinkles, weight gain, fat distribution around the body, and more.
Like weight gain and nerd transformations, age progression can be part of a punishment/humiliation story. It can also be about beautification and empowerment, depending on the perspective of the writer. Or it may simply happen, independent of any overarching reason.
The story that introduced me to the genre is "Spider Bites" by greinskyn, and it has some great visuals and a fascinating set up. Generally I'll prefer a fantastic concept told adequately to an adequate concept told fantastically, and this story is certainly the former.
Ones of my personal favorite age progression stories is "Graduate to Maturity" by Kronostar. This story is about empowerment and, in a rare twist, is a voluntary transformation. The voluntary change motif is one that I enjoy in part because I feel it makes a change more consequential if it is something chosen, rather than merely received.
I'd say the biggest dividing line in age progression stories is how old the person usually gets. My unscientific observations suggest that about two thirds of those interested prefer to see characters reach ages of around 40-60, in other words, middle aged, while the rest want characters to become very elderly, usually the older the better. My interest lies with the first camp.
One aspect of age progression that I enjoy in particular I feel has been generally neglected - the ability of age progression to imbalance the power dynamic between men and women. A woman who suddenly becomes an older woman might be economically more successful, have more sexual experience, more confidence in what she wants, or many other traits which are traditionally associated with dominance in a relationship.