The Remington Model 51 was introduced in 1918 in .380 caliber. Unusually, they did not make it in .32 until 1921 and there were only about 10,000 .32s out of a total production of 64,000+. They were manufactured only until 1927, but a few were assembled from parts on hand until 1934. Compare with Colt, who made over 400,000 guns in .32 and .380 from 1903 until 1946.
The design was by John D. Pedersen. It is a delayed blowback action, with a separate breechblock working inside the slide.
The M51 was never official military issue, even if Gen Patton had one. The anchor stamp on many of them was a Remington company inspector's mark and had nothing to do with the Navy. However, the Navy HAD looked hard at the M53, the same action in .45 cal but with exposed hammer. If we had not entered WW I, they might have adopted it.
It is a well made gun of unusual design. As a shooter it is noted for its well-shaped grips, good sights for such a small gun, and a tough trigger pull.
Looking at the slide pull serrations (original style) your gun was probably made in 1921 or earlier.
Although they are much less common than Colts, they are worth no more, if as much. The latest Gun List has one at 85% blue, exc bore and grips, for $275. A .32 would be more.
No reason not to clean and shoot it. Takedown is not obvious, if you can't find instructions, let me know and I will key them in. I don't think that .380 ball has been souped up beyond the original pressure and velocity. I wouldn't push it with Cor-bons, it probably wouldn't feed the hollowpoints anyway. Mine wouldn't, that was a big thing to me at the time, so I traded it. Wish I hadn't. I have replaced the Colt .32 I had in those days, but not the Remington.