Only the thumbnail is attached, so i can't give a positive ID. It's a top break with an exposed hammer. These came in .32 S&W (NOT .32 S&W long) or .38 S&W (NOT .38 S&W Special).
Assuming .32 S&W, then you have a .32 Double Action 4th Model manufactured between 1883 & 1898 in the serial range 43406 to 209301. The trigger guard shape is correct for that model. The grips appear to be are aftermarket, if real ivory or mother of pearl they may add to the value, if they are real ivory or mother of pearl & have a S&W medallion they will add to the value as these were a special order item.
This is a black powder only gun. Don't shoot modern factory ammo through it unless you want to see it wear out quickly. Although modern ammo is set to the same pressures as the old black powder loads, smokeless powder has a much faster pressure peak, putting extra strain on the steel (iron really) of the gun.
It also lacks any form of hammer block safety, so treat it as a four shooter, lewaving the chamber under the hammer empty.
If the caliber is ,38 S&W then it is a .38 Double Action 3rd Model manufactured between 1884 & 1895. The issues with black powder & drop safet are the same.
Both models are classified as antiques by the Federal government as they were made before 1998.