Stabilizer is another name for CYA (or conditioner). It basically keeps the sun from burning off all of the clorine during the day. You want to be between 30-50ppm.

No, the clarifier is not the same thing. A clarifier helps to bind really fine particles into larger ones so your filter can catch them. From what I have been reading on the forum the clarifiers are generally not necessary for normal pool maintenance - but I am no expert