The inhabitants of towns and cities send mail to each other about various things. Mail that has to be sent any real distance (i.e. across or between towns) is left at the local station (should it be accepting mail) for a transport company to collect and deliver.
Your company Headquarters also generates a reasonable amount of mail; presumably from the advertising materials, bribes and various other items of business correspondence you send out in the course of your OpenTTD game. Naturally it also accepts mail deliveries from your customers and suppliers.
You can read more about Mail at Wikipedia:Mail.