1. In USA and Great Britain these tales are known as Arabian Nights (sometimes Arabian Nights Entertainment) but many of them have nothing to do with Arab countries. The collection consists of fairy tales, fables, riddles and verses from large area of Asia and Africa, with India probably being the biggest contributor. The main story which serves to present all the others, for instance is going on in Persia.
2. In Europe Arabian Nights are mostly known as 1001 Nights. Many people believe Thousand and one is the number of the tales in the collection but this is also not true. 1001 Nights only gives a time frame in which stories are told by Scheherazade to her husband. He wants to kill her as all his previous wives after the first wedding night but decides to postpone the execution because he wanted to know how story ends (and she always starts the next before he notices the first one is finished).
There is no definite answer to the question "How many stories are in 1001 Nights?". There are versions with only a dozen tales and versions with several hundreds available. We also have a problem to identify which are the 'original' tales, which came into collection from other sources and which are not a part of folklore at all.
And of course pretty complicated narration with frames, unfinished stories and two or three variations of many of the stories don't help either...