One of the websites listed below has a large collection of the stories. However, I do not believe it is possible to get all 1,001 stories. There are many tales that have been lost over time.
There is a book for sale entitled, The Arabian Nights: Tales from a Thousand and One Nights, printed by Modern Library Classics in paperback, but it does not have 1,001 tales. However, some of the tales start with one story and then they have three more tales included within the original that must be told before the original story is concluded. Because of this, it is difficult to say exactly how many stories the book contains.
The collection of stories is primarily of Asian and Arabic origin. They were very old and had been passed down in the oral tradition (by word-of-mouth) by "scops" or "professional storytellers" for a very long time, but not recorded on paper. This would account for some stories being lost, while others may appear with different details, changing over time among the multitude of people who told the tales.
Originally, The Arabian Nights, also called A Thousand and One Nights, was made up of "legends, fairytales, romances," and narratives from a variety of "folk traditions." The stories were originally "born" in a collection of geographical locales, and from many different historical time periods. In the 18th Century, they reached the West and quickly became very popular. Sir Richard Burton was said to have collected many tales in a dangerous manner, traveling into towns where outsiders were not permitted— escaping detection and possible execution.
There is a reason it is difficult to find all of the original tales. The way the stories were passed along before they were written down is one major factor. The other is the loss of physical manuscripts once the stories were recorded.
The first identifiable written version of The Arabian Nights is a book of Persian tales called Hazar Afsanah (A Thousand Legends, written between 225 and 250)...
Somewhere around 850, the stories were translated—into Arabic. Al-Mas'oodi, a writer of the 10th Century, refers in his writings to the Arabic translation—but since then, the translation has been lost.
There is no definitive Arabic textual source of the work, but there are a number of surviving manuscripts containing many of the stories.
So while there are many stories still available in a variety of books and surviving manuscripts, getting all of the tales seems to be an impossible task. The tales are timeless, still popular today, all over the world. As to summaries, eNotes offers Gale summaries of a few of the tales.