The ListBox is basically irrelevant. If you want to persist data between sessions then you have to save it to some external store, e.g. a file or database.
If all you want is a list of Strings then the simplest option may be to add a StringCollection on the Settings page of the project properties. The contents of that collection will be automatically saved to the application config file at shutdown and automatically loaded at startup. You can populate the ListBox from that collection at startup and then populate the collection from the ListBox at shutdown.
Note that, in code, you access application settings via My.Settings.