If you want any text within a note to be treated as private, you should include it in square brackets, like this: [[secret text]]. This is called private text or a private note. You can include the entire contents of a note within the doubled square brackets, or any part of it. It may be just a word or phrase that you wish to treat as private (e.g. "he limped as a result of a [[self-inflicted]] war wound"). Ordinarily, private notes are excluded from reports and books, or from pages in websites or family tree CDs and DVDs, when these are based on reports. But they can be included if you wish. The relevant option is usually to be found on the main page of options for the report or book in question. If you choose to include private notes, you can also choose whether or not to include the double square brackets around them.
Private notes can be entered into any Note field, and within any note field you can have as many private notes as you like. A 'note field' for these purposes means any field labelled 'Note' (it doesn't include Named List notes though - see below). You can however also include private notes in a small number of other fields, which are not labelled 'Note' - such as "Address" fields, or the "Text from Source" field on Source Citations and Source records. If a field does not allow multiple lines of text to be entered into it, it does not support private notes. If you double-click on a field that supports multiple lines of text, a Memo window will open to make it easier to enter text on multiple lines. You cannot enter multiple lines of text into Name fields, Date fields, Age fields etc, so none of these support private notes. Another example of a field which does not allow multiple lines of text is the "Where within Source" field on Source Citations. Although superficially similar to the "Text From Source" field, you can enter multiple lines of text into the latter, but not into the former. Nothing will prevent you entering text enclosed in doubled square brackets in the "Where Within Source" field; but if you do so, it will not be treated as a private note and will be displayed as entered in reports and books, regardless of options chosen.
Private notes are not used with Named Lists. You can enter a note for a Named List and/or for each item in a Named List. And there is nothing to stop you entering text in doubled square brackets in these notes; but the text will not be treated as a private note.
If you wish to delete all private notes from a GEDCOM file, the Split Tree Helper tool (accessible from the File menu) can be used to do that.
If you export a GEDCOM file, private notes are excluded by default - but you can choose to include them if you wish (see the Export Gedcom File Options dialog).