DNIS is an acronym for “Dialed Number Identification Service”. A DNIS is a service offered by telcos which allows the identification of the phone number that a caller dialed.
For example, if you are calling a customer service call center from your home phone, the DNIS service allows the call center to see which number you dialed.
Here’s an example.
United Airlines has several customer service phone numbers. They publish their main customer service number on their website for everyone to see. They let some of their most frequent fliers know about another (more private) customer service phone number that is staffed by more skilled agents.
The two phone numbers both get you to United customer service, but the treatment you receive if you call the private number is totally different than if you call the public phone number.
DNIS is related to ANI in that ANI tells the called party the number of the phone that originated the call, while DNIS tells the number that was originally dialed (which may differ from the end phone number due to call forwarding).