A half pad is extra padding sometimes put under English saddles - some people believe it makes the horse more comfortable.
The half pad goes over the horse's back under the saddle, but only covers the "seat" of the saddle and not the flap. They're traditionally made of sheepskin or cotton, but these days you can also get memory foam. Sadly, some people use them so they can continue to use a saddle that doesn't fit properly. They are, though, helpful for horses with more sensitive backs.