Some parts of the documentation state that an “Ad Group” is simply a “Campaign” that contains more than one “Line Item”: https://dev.twitter.com/ads/overview/ad-groups-faq
Other parts, however, state that “Ad Group” is equivalent to “Line Item” and, in this context, campaigns with more than one “Line Item” are called “Ad Group Campaigns”: check the Line Item paragraph at https://dev.twitter.com/ads/tutorials/hierarchy-and-terminology
To me, the first definition (“Ad Group” = “Campaign with more than one Line Item”) makes much more sense.
There is no clear equivalent of an ad in the Ads API hierarchy. To some people and documents, a “Line Item” is an ad (this stance would be coherent with the first definition of “Ad Group”). To others, the “Promoted Tweet” or “Promoted Account” within the “Line Item” would be the ad (in line with the second definition).
I have never heard “ad-set” in the context of Twitter Ads. That’s a Facebook concept.
Maybe someone else can provide an authoritative explanation.