Word and character count is shown in the status bar, and is kept up to date as you edit.
If you want to count only some text of your document, select the text.
To display extended statistics such as count of characters without spaces, double click the word count in the status bar, or choose Tools - Word Count.
How does LibreOffice count words?
In general, every string of characters between two spaces is a word. Dashes, tabs, line breaks, and paragraph breaks are word limits, too.
Words with always visible hyphens, as in plug-in, add-on, user/config, are counted as one word each.
The words can be a mix of letters, numbers, and special characters. So the following text counts as four words: abc123 1.23 "$" http://www.example.com.
To add a custom character to be considered as the word limit, choose LibreOffice - PreferencesTools - Options - LibreOffice Writer - General and add the character into the Additional separators field.
To get some more statistics about the document, choose File - Properties - Statistics.