Mouthing means getting a horse accustomed to having a bit in its mouth.
These days, traditional mouthing has gone out of fashion - but I admit I'm still a proponent of it. Traditional mouthing means putting a bit and bridle on the horse in a stall and leaving it loose with the bridle on, with no reins attached. The horse is left at first for only a couple of minutes, with the time eventually built up to an hour (at most). Often the bit used is a special "mouthing bit" which has rollers or metal keys on it, which encourage the horse to play with it. Molasses or unsweetened apple sauce may be put on the bit.
Mouthing is also used for the process of lunging a horse with loose, elastic side reins to start accustoming it to pressure on the mouth.