Fixing the SolidWorks error takes about 20 minutes, and is related to a Windows update in July that changed security settings for installing programs.
In order for this to work, SolidWorks has to be completely installed through the .ISO we all downloaded, using the HTML launcher in the file. Registries as well as all parts of the original install will need to be removed.
Next, a new registry needs to be temporarily added in order to bypass the new security feature. Using regedit, follow the file pathway HKEY_LOCAL_MACHINE\SOFTWARE\Policies\Microsoft\Windows\Installer (on many machines using Windows 10, "Installer" will not exist and will have to be created by right-clicking the windows folder and making a new directory).
Under Installer, create a new DWord Value (under edit > new) and name it RemappedElevatedProxiesPolicy, right click it to Modify and change the value to 1
Reinstall SolidWorks using the ISO. You can either leave the new registry as it is or change it back to zero, however I'm not sure whether or not resetting it to zero will create new errors.
According to the support forums, the problem is going to become more pervasive and affect a lot of more advanced functions of SolidWorks, as well as cause it to slow significantly on more advanced projects, so everyone with a bad install needs to change it as soon as they can.