The Colt 32 automatic says "32 Rimless Smokeless" on the side. That name never caught on. The ammunition that pistol uses is known by several different names today:
32 ACP (ACP means "Automatic Colt Pistol")
32 Automatic
7.65mm Browning (This pistol was designed by John Browning, who designed many other famous guns, including the US Army Model 1911 45 caliber automatic pistol)
I have sometimes been able to find this ammunition at Wal Mart, but I have not looked in a long time.
With a gun you are unfamiliar with, it is best to start off with "full metal jacket" (FMJ) ammunition, because that has the least chance of causing jams. The standard FMJ bullet for 32 Automatic weighs 71 grains, IIRC. Fortunately, this is the cheapest kind of pistol ammunition.
"Hollow point" ammunition has better stopping power (although that's a whole big argument), but it did not become popular until long after the Colt 1903 went out of production. It will not harm it, but it may cause jams. There is no way to find out besides trying some.
If you could put up pictures of the gun, people here can tell you more about it. There were several variations of the Colt 1903, although the basics were all the same.
There is a good article about them, with instructions for taking them apart, here:
http://unblinkingeye.com/Guns/1903C/1903c.html