The ledger board is attached to the inside bottom edge of the side rails. It is what supports the slats (or alternately plywood sheets) that support the mattress, that were discussed in the previous bed thread. It is usually about 1" square, and attached with screws, and maybe also glue.
If you guess wrong on the thickness of the mattress, about the worst that will happen is it will either be difficult to access the bottom row of cubbies, or your pillows with slip onto the floor between the mattress and the headboard. Of course , if ithe mattress ends up too wide, you have to start over. If it ends up too long, you will need to replace the side rails. If shorter or narrower than optimal, it will still be supported adequately by the slats.