The information in this article is from game version 1.4.6 on Steam.
The New Gods are a pantheon of gods themed after modern obsessions. They serve as the bosses for floors 1-4 and have two different difficulties each, the easier variant is encountered on floors 1 + 2 while the harder variant can be found at the end of floors 3 + 4. The Managers are the higher tier bosses and will be encountered from floor 5 and onwards.
In general, every new god fight consists of two phases. Phase 1 will usually last until the god's hitpoints reach 40-60% while phase 2 then lasts until the boss is defeated. Phase 2 usually changes and/or speeds up the bosses attack patterns from phase 1.
New gods will always drop one item on death except Meta, the God of Bittercoin on Floor-?, who always drop two items.