LDN stands for Local Directory Number. An LDN can only be attached to the LCRG (on AS/AX/DX units) or in the PBX trunk group of older A400/A800. Quite simply, they are the phone numbers that people call to the unit. For example, if you had a phone connected to the FXS/PBX ports of an A800 or AS/AX unit and you wanted to assign it a phone number of 5551000 so that anytime someone called 5551000 and it came in to the Tenor it would ring the phone, you would configure 5551000 as the LDN for the trunk group where the phone is connected.
For PSTN trunk groups, you would use hop-offs (HND) also called LAMs that allow you to strip and add digits.
As for public and private, that could get very confusing. There are many documents that try to explain this concept and many people may offer their opinions on this, but quite simply a public number is a phone number that can be routed over the “real” phone network. It must follow a specific format/number of digits. A private number is one that has no format and is made up by the users. A private number cannot be routed by the public phone network. An example of public and private numbers is if you are in an office with a PBX. You have public phone numbers where your customers and others can call from outside your company to you. Yet you may be able to dial a 4 digit extension to call the person in the next office over from you. The 4 digit extension dial is a form of private dialing.