You need a GROUP BY, and as this isn't MS-Access, I don't think that the #..#'s are going to work:
SELECT Booking.Lesson_Date, Lesson_Type.Lesson_Desc, COUNT (Booking.Lesson_CatId) AS TotalTestAndSessions
FROM Session_Information
INNER JOIN Lesson_Type ON Lesson_Type.Lesson_TypeId = Session_Information.Lesson_TypeId
INNER JOIN Booking ON Booking.Booking_Id = Session_Information.Booking_Id
WHERE (((Booking.Lesson_Date) IS NOT NULL) AND (Booking.Lesson_Date BETWEEN '01-Oct-2008' AND '31-Oct-2008'))
GROUP BY Booking.Lesson_Date, Lesson_Type.Lesson_Desc
Also, there's no need to nest INNER JOINS, you can always just connect them serially.