direct number - when called reaches a specific user extension. Each user gets 1 direct number which is tied to its license(this is called a DL- digital line). “Additional local numbers(ALNs)” can also be assigned to the user. 
All ALNs are direct numbers. 
Not all direct numbers are ALNs (for example DLs are not a ALNs, but they are still direct numbers)
One way to tell ALNs apart from DLs is that DLs have greyed out checkboxes - bulk actions are not allowed for them. In the attached pic 1 user extensions has 3 direct numbers. The greyed one is a DL the other ones are ALNs.

It’s possible to assign direct numbers(ALNs) to other extensions besides users.


company number - when called reaches the auto-receptionist for the site. From the auto-receptionist the company number might be routed to one of the following:
- a user extension (then it behaves the same as a direct number)
- a call queue 
- other stuff

softphone - web app, mobile app, or desktop app. As opposed to a hardphone - which is a deskphone.

 

This is great and very helpful, thank you