The different settings (Options dialog etc.) are recorded in various ".xml" files in the Simple DNS Plus application data directory:

On Windows Vista / Server 2008 and later, this location is typically:
C:\ProgramData\JH Software\Simple DNS Plus
On earlier Windows versions:
C:\Documents and Settings\All Users\Application Data\JH Software\Simple DNS Plus

The actual zone data files (zones and records) are stored in a separate directory as specified in the Options dialog / DNS / Data Files section:

To backup: Copy all ".xml" files, and ALL files in the data directory.

To restore the settings and data on a new or reinstalled computer: Install a fresh copy of Simple DNS Plus and copy the backup files back to their original location.