The TimeSprite Database is located in the Local Application Data folder in the TimeSprite subdirectory. This is typically in a location like:
C:\Documents and Settings\<username>\Local Settings\Application Data\TimeSprite
Note: some of the folders in this path are hidden by default, so to find it you need to select "Show hidden files and folders" in the Windows folder options.
If you need to move your data to a new PC, simply move the contents of this folder to the equivalent location on the new computer.
You can change the location of the database if necessary.