OS X install does not install OS 9. You can install it separately - UNLESS you have a new powerbook or PowerMac, in which case you cannot install OS 9 (or earlier). The machine simply won't let you do it.
"Classic" is part of the OS X install however. This "walks like a duck, quacks like a duck" as far as looking and working like OS 9 but it is not a separate OS 9 installation: and only apps designed for it will work in classic. You may have heard the term "carbon" or "carbonized". If an app is designed to run in "classic mode" it is said to be "carbonized."
This classic environment means that virtually all apps designed for OS 8.5 and earlier will not run in "classic mode."