A posted order is a list of who goes when in the class, determined by the show office. Posted orders are generally used in the highest priority ring at the horse show. How faithfully the in-gate follows that posted order will depend on the horse show and on the particular class. You won't be able to move around in a big grand prix, but in smaller classes you will sometimes be able to change your order if you have a good reason to (trainer conflict, rider conflict, etc.). If you absolutely will not be able to be in the ring for your spot, talk to the in-gate ahead of time. The worst thing that you can do is to just not show up until later.
Classes like grand-prix, mini-prix, medals and hunter classics or derbies are generally run as posted order.
Sign-in orders are more casual orders of go that are created by the in-gate person. Before your class starts, you will need to go speak to him or her to put yourself in the order. Riders can choose any spot they want out of whichever ones are left (it pays to put yourself in the order early in the day so that most of the spots will be open). There is more flexibility in this type of order since spots might need to be opened up for trainer conflicts, so you might not end up being called into the ring exactly where you signed up, but it should be pretty close.
How do you know whether your class will be run as a posted order or as a sign-in? The best way to find out is to ask the in-gate person directly. There are, however, some clues that can give you a good idea before you ask. Check the exhibitor numbers on the class list (it should be posted at the in-gate and at the show office). If those numbers are in increasing order, chances are that you're looking at a class list rather than a posted order. If those numbers are random, chances are that it's a posted order. Sometimes the office will make the class list look like a posted order anyway, just in case the show wants to run it that way, so it's a good idea to check with the in-gate to make sure that you don't need to sign in. If there is a signature on the class list, it is most likely a posted order.
If you're riding in the hunters, you should familiarize yourself with some terminology before signing in. Because hunter classes are generally run in groups of two or three over fences at the same time, there is a difference between the number of 'trips' and the number of 'horses'. A 'trip' is a single class, whereas a 'horse' is two or three classes (depending on the schedule). Ask the in-gate specifically whether you are putting yourself in the order at a number of trips or at a number of horses so that you can plan your time accordingly.
Jumpers don't need to worry about this since only one class is run at a time, therefore the number of rides is the same as the number of horses.