The add/scratch sheet
Somewhere in the show office, you should find a pad of copy paper that will probably have the title 'Add/Scratch Form' written on it. It is a very short form, usually divided into an add column and a scratch column, with about five fields: the date, the horse's name, the horse's number, the rider's name and the class number.
The date is on there to determine whether or not the horse will be charged a late entry fee/scratch fee (it's not really important to fill that field out if you will be handing it directly to a secretary).
The horse's name and number are obviously important in identifying the horse. Make sure that you know the horse's number before entering the office in order to make life easier for the office staff.
The rider's name is important if you want to make sure that the correct rider is announced and credited with any results. It's also very important if you are entering any equitation classes.
The class number is, of course, one of the most important parts! The office should have extra copies of the show schedule available that you can check to find the correct class number.
Once you're done filling the form out and have signed it, hand it to the show secretary or place it in the designated basket If you are adding or scratching the day of the class, you will be handed back a copy of the add/scratch form to bring to the in-gate so that you can be added to the class list there.
You will usually be charged an extra fee if you add or scratch the day of the class. If you add or scratch before the office closes the day before your class, you won't be charged any more than the class fee itself.
The trainer form
If you are adding or scratching multiple horses, individual add forms can be time-consuming. In such a case, ask the secretary if it would be possible to print off a trainer add form for you. These forms list all of the horses that are at the show with a particular trainer, complete with their show numbers and which classes they are already entered in. All that you will need to do to add or scratch is write the class number in the proper column (add or scratch) beside the correct rider's name (all of the riders from the original entry form should be listed). Sign it, hand it to the secretary and you're done!