Can I install programs on an external drive?

If when you set it up a program allows you to choose a destination drive to install onto, it shouldn't matter if that drive is C:, your Windows drive, or some other hard drive internal or external. For the most part they're all just drives, and the fact that the setup program gives you the option means there shouldn't be an issue.

For the record, if the setup program doesn't give you a choice then it may be an issue and I'd avoid trying to circumvent it.

Now, many external drives are removable. USB drives, for example, can be removed from your computer either while it's not running, or after clicking a "safely remove hardware" for the device. In the later case, if a program on your removable drive is running, "safely remove" may fail. All files on the external drive must not be in use, and a program running from the drive is definitely "in use". You'll need to make sure that all programs installed on the external drive are shut down before attempting to remove the drive.

Now, that sounds simple enough but more complex packages often install software that runs automatically when you start your machine, either in the form of explicit start-up menu items, or as system services. You may not think you're running any software that's installed on that external drive, but in reality it may have installed something that is. You'll need to figure out what that is, and take steps to disable it, or shut it down before you remove the drive.

Along those same lines, if the drive is not installed when you boot your system, you may get error messages on start up if a prior installation of software onto that drive also added startup items. Without the drive installed, those startup items may not be found, and errors may result. Menu items and shortcuts relating to the software installed on the external drive may also revert to Windows defaults until some time after the drive is reinstalled.

Finally, don't expect application portability by installing it to an external drive. By that I mean that installing an application to an external drive, and then taking that drive to a different computer and expecting the application to work on that machine. This may work for simpler applications, but setting up a complex application is as much about setting up the Windows registry as it is about just placing files. The registry will have only been set up on the first machine. Theoretically it might be possible to set up the software on machine A, then set it up again on machine B if the drive letters are identical, but I'd expect that to be highly error prone. It's also quite possibly against many applications licenses or terms of use since you're technically installing it on two machines.

So there are a number of issues related to removable media, but if you're just installing your software to a second hard drive, and it just happens to be an external drive, I wouldn't expect that scenario to be an issue for you.