By my understanding, AFA's are basically a blanket term for any fee arrangement which is not a classic all-hours-billed or fixed fee model - for example, a client having an 'account' of X number of hours per year/ block hours discounted/ etc etc.
Fixed Fee and AFA models will be far more common at major City firms; CFA's tend to be more for private client work, such as employment discrimination or personal injury and other claims in tort.
Again, the caveat is that the above is only by my understanding, so do feel free to point out if I'm under a misapprehension.
Damages-based agreements are also something that can sometimes occur, but typically they are used by firms that do a lot of high-value PI (e.g. spinal injury) or product liability work.