If I supply a 12V Output Linear Voltage Regulator with 9V, what will happen
It depends on the design. Most typically you will get an output voltage somewhat below the input voltage, so perhaps 7 or 8 V. In a "fancier" design the output might be totally suppressed by a brown-out detection circuit. Or, really, anything can happen because operation in these conditions is not specified in the datasheet.
firstly if the regulator's dropout voltage is under 9V and secondly if it is not
The drop-out voltage is normally specified as a difference between the input voltage and the output voltage. So if the drop-out voltage is 9 V, you will need to provide at least 21 V at the input in order to get 12 V at the output.
But 9 V is a very high value for drop-out voltage. 3 V or so (requiring ~15 V at the input for 12 V output) was common in old designs, and 0.5 - 1 V (requiring 12.5 or 13 V at the input for 12 V output) is common in new, "low drop-out", designs. Even lower drop-out voltages may be available, but watch out that the quality of regulation may drop as your input-output difference voltage nears the drop-out limit.