CorsixTH is an open source clone (a game engine recreation) of Theme Hospital that aims to get the game running on modern platforms, since the original release was only developed for DOS/Windows 9x (Origin and GOG use the DOSBox compatibility layer). You can download the installer from GitHub here and point it to the location of the original game. Once you have this installed, you won't need to run HOSPITAL.EXE anymore; if you choose to put CorsixTH as a shortcut on the desktop, it should effectively replace the original game. Also note that you usually get better results using a "clean" version of the original, i.e. one that isn't patched.
While the recreation's stables have every feature implemented, the update cycle is very slow. The latest fixes and features are on the development branch, which doesn't have builds and doesn't receive as much testing. However, it may solve bugs you encounter with the current stable. The GitHub repository's wiki has a page detailing how you can compile it for your system.
Getting the game to run at all on modern machines is difficult due to the age of the game. However, there are several steps you can follow that may get it working. It is suggested to check if the game works after each paragraphs as different users will have different settings.
The first is to transfer the HOSP folder from your CD copy onto your storage device, making sure to save the files close to the root of the drive (the entire folder location needing to be under 8 characters, e.g. C:\ThemeH\. You will also need to edit the configuration file to allow saves to work (see Game data).
Now change your EXE compatibly settings (which may depend on what OS you are using). Right click HOSPITAL.EXE and click properties. going over to Compatibility and choose Windows 95. Then, under Settings check Run in 256 colours, Run in 640 x 480 screen resolution as well as Run this program as an administrator. Fiddle around with these options for the one that suits you if you know what you are doing.
There is also a patched version available, which you can download here. Copy and paste it over to your HOSP folder (replacing any existing files, remembering to edit the configuration file again if you replace that too). If this now gives you mouse lag, see Patches.
If none of those steps work for you, then you will have to use CorsixTH. (see "CorsixTH").
Mouse Lag Patch - this sets the CPU affinity to 3 instead of 1, though you may need to fiddle with this number. Save this file where your HOSP folder is and run it instead of Hospital.exe.
 Follow the instructions to fix the music in the GOG version