You have a fairly early S&W 357 Magnum, known by collectors as the Registered Magnum. Based on the REG number it is a fairly early gun, probably 1935. I know that General George S. Patton's gun is REG 506 and was ordered in September of 1935, so yours should be at the end of 35 or early 36.
If you go to the top of the Hand Ejectors: 1896 to 1961 you will find a search feature. Just type in "Registered Magnum" and you will have days of reading.
As I said before, this is the Holy Grail of Smith and Wesson Collectors. These guns were all basically custom guns with barrel lengths between 3.5 and 8.75 inches in 1/4 inch increments. I believe the six inch barrel was the most common.
Like most S&W revolvers, condition and provenance greatly affect value. The photos look good, but I'm no appraiser as to condition and value. It does appear to have the original finish, which enhances value.
We, here, love photos and stories. Tell us about how you came into possession and detailed photos will help others to give approximate condition and value.
Again, welcome to the forum and you have made a grand entrance and have a gun that many aspire to.
Edited to add: This link will take you to a thread here on the Forum which goes into detail about the Registered Magnum
What Is A Registered Magnum