The term “ANI” refers to “automatic number identification”, a feature of telephone networks which allows the origination phone number to be identified on toll calls. ANI's are traditionally used for billing purposes (to tally up the amount of minutes of usage a particular phone number should be billed for in a given period of time).
ANI is different than caller ID in that a phone number may be configured to block it's caller ID, but blocking the caller ID does not block the ANI (as the ANI must be tracked for billing purposes).
The advent of VOIP calling services has changed the way ANI's are used. Some VOIP services present a call using a non-working ANI making the use of ANI less reliable.
ANI is related to DNIS in that DNIS tells the number that was originally dialed (which may differ from the end phone number due to call forwarding), while ANI tells the number of the phone setting up the connection.