I've been handloading and shooting the .44Colt for several years now. Originally, it was conceived as the largest metallic cartridge that would physically fit in the platform. It was loaded with a heeled, hollow-based bullet to swell up in a percussion .44's .45 caliber bore. Modern .44Colt, which is what I assume you're asking about based on past discussions, is basically a slightly shorter .44Spl with a smaller rim. It uses the same .430" cast bullets as the .44Spl and Mag. I use Titegroup exclusively with a 200gr cast RNFP, 240gr cast SWC or 240gr swaged Hornady SWC-HP.
You can shoot .44Colt in any .44Spl or .44Mag sixgun. However, your .44Colt might not be. I have two Cimarron .44Colts, an 1871-1872 Open Top and a Richards Type II 1860 cartridge conversion. Both are marked "CAL. .44 COLT". The Open Top has proper .44Colt chambers but the 1860 has .44Spl chambers.
I got my loading data from John Taffin's writings, Hodgdon and I believe LoadData.com.
Far as I know, everything above also applies to the modern .38Colt.