The Army Special was made from 1908 to about 1927. At that time they changed the name (but not the gun) to Official Police, because it was clear the Army was no longer interested in .38 revolvers as a standard issue sidearm. As the Official Police it continued in production until around 1970 when it was replaced with a new model.
It was offered in calibers .32-20, .38 Special and .41 Colt. Those chambered in .38 Special can also fire the slightly shorter .38 Long Colt, but there is no good reason one should do so. I have no idea why your Father had the older (and obsolete) Colt cartridges, but now they might be collector's items.
Be sure there is no leading at the front of the chambers, and if there is scrub it out. Then use regular .38 Special rounds. Limited use of Plus-P ammunition won't hurt the gun, but for general plinking & playing use standard velocity cartridges.