Smith & Wesson info.

.32-20 Hand Ejector Model of 1905 4th Change manufactured between 1915 & 1940 in the serial range 65701 to 144684.
Built on the K frame, your gun has a heat treated cylinder so it is safe to shoot with lead loads. Check for a bulge in the barrel, many owners made the mistake of shooting the rifle only hi-speed loads through these guns. The jacket would strip off in the barrel and the next round fired would bulge the barrel.
Your gun predates the current positive internal hammer block safety introduced during World War II. If dropped it could fire, so leave the chamber under the hammer empty.

Hand Ejector Model of 1905 4th Change

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That is the model/what its called. Smith & Wesson did not have specific numbered models before 1957.
The caliber is .32-20 Winchester and you would shoot .32-20 Winchester in it. This ammo is only common with cowboy action shooting and can be hard to come by/expensive.

A call to S&W will get you a date on when it left the factory.

Radagast pin pointed about as well as can be done without the $50.00 letter in post #2. Your 12XXXX gun would have be shipped late in the period of production, maybe even the last year.

Cajunman89:
No, they couldn't give you the date over the phone as the records from the 1930s are still on paper and are no longer maintained by S&W. For $50 Roy Jinks the factory historian will go back through the original paper shipping records by hand until he finds your gun, hence the $50 charge.
Prior to World War I the .32-20 was shipping at a rate of about 4000 guns a year. Allowing for the effects of the great depression on sales, my guess is late 1920s to early 1930s.

Go to the S&W website and you can download a current revolver manual. Stay away from removing the sideplate or cylinder. Leave that to a good gunsmith if it ever has to be done.

Here is a link to a Model 10 manual.

http://stevespages.com/pdf/s&w_10.pdf

Be aware that there have been some 17 design changes since your gun was produced, so the modern S&W parts list will not match. The action is not the same. I'd leave working on it to a gunsmith with experience with pre-WWII S&Ws.

the standard catalog shows that serial number being a 1st change

Of course, as they have some incorrect info so do not take this as fact

Nope. They would simply show it to be a 4th change variant, which we already know from the serial range.
I would pick it to be from the mid 1920s. Over a 13 month period in 1923-24 dome 15000 guns shipped, reaching serial number 125183. If I had to guess I would say 1924 or 1925.