I am affil for sj & dr. All it means is that you've paid money to join British Dressage or the British Showjumping Association. You have to pay separately to reg the horse & the rider & you get a reg no to put on your entry forms.
For BSJA, the difference with unaff is that the courses can be better quality (& up to height!!) For BD, the judging tends to be more consistent. If you win stuff BSJA, the money is logged on the horse's permanent record & once you've won a certain amount, you can't do the smaller classes. For BD if you score over 60% at novice & above (not prelim) you get points that go on the horse's permanent record & once the horse has got more than a certain no of points, you can't do the lower classes.
There are whole rule books of ins & outs, but that's the bare essentials!!
(You can affil for eventing too but I don't know anything about that, apart from that it's £££££££)
affiliated is when you are affiliated to a certain competitive organisation for example British eventing - starts at 90cm with basic dressage, BSJA - starts at 80cm i think and British Dressage - starts at prelim