Obadiah is much more specific than Zephaniah – He is prophesying against Edom.
This book, too, is structured with the hoped for future near the end of the book.
The circumstances of the writing appear to be following a raid by Philistines and Arabs, accompanied by the revolting (at the time!) Edomites.
II Chronicles 21: 8-10, 16-17. This would place the authorship in the time of King Jehoram. Other views place the book later, even as late as 585 and the deportation of the Jews to Babylon under the Zedekiah. So the book is either from the 9th century or the 6th. I tend to think it is the 9th – we can discuss why, or argue why not on Sunday.
You can read the genealogy of the Edomites in Genesis 36 – they are the descendants of Esau.
A brief history of the dealings between Israel and Edom:
Numbers 20: 14-21
2 Samuel 8: 13-14
II Chronicles 21: 8-10, 16-17 (as above)
II Chronicles 25
II Chronicles 28: 16-21
II Chronicles 36: 11-21