I'm sure i understand what you are trying to accomplish.
you want to filter the recordset to return records where the people.firstname and people.lastname have a specific value?
if so, that is not done through a join, that is done using the where clause:
SELECT inbox.inboxID, inbox.newUserID, inbox.applicationID, inbox.inboxDate, people.peopleID, people.firstname, people.lastname, people.salutationID, application.applicationID, application.peopleID
LEFT JOIN people ON inbox.newUserID=people.peopleID
LEFT JOIN application ON inbox.appilicationID=application.applicationID
WHERE people.firstname = "john" AND people.lastname = "doe"