You have a S&W Safety Hammerless Double Action .38 S&W. They very popular and were sold from 1887 til 1940! Nearly half a million .32s and .38s were made in five models which differed mostly in the top latch.
Not many probems, they were apparently a better gun than the exposed hammer S&W top-breaks as made from 1880 til 1911.
Of course they are an old design of old or very old manufacture. You are pretty much on your own to shoot it. But I shoot my even older .38 Single Action with factory .38 S&W ammo, and many Cowboy shooters do likewise.
Book value $200 and up, but not way up, depending on condition and exact model.