The 9V regulator in this datasheet has a 2V dropout voltage. However, in the electrical characteristics on Page 9, the datasheet says that you need 15V input to get a 9V output. Does that mean I can't get 9V out with a 12V input? Or am I interpreting the datasheet wrong?
Yes, this regulator will give you 9V from a 12V input. The dropout voltage is indeed 2V.
The heading on page 9 is just telling you the test conditions that were used to characterize the part and get the data shown in the table. It's not practical to specify a device at every possible operating point, so the manufacturer chooses some typical conditions and measures the device parameters under those conditions.
If you look at the quiescent current specification, for example, you will see a maximum value of 6mA. The section on quiescent current change gives you an idea of how the quiescent current will vary for a few different cases: varying input with a smaller load, varying input but held at room temperature, varying the load current with constant 15V input.