Yes, you can use the "Analog Input" pins as digital I/O.
So you have 14-2(for USB) = 12 plus 6 = 18 Digital I/Os.
Look at the connectors on the board you bought. There are two:
One goes to Arduino: 5V and GND, and the 16 relay signals to the I/Os. This drives only the logic part of the board, not the relay coils.
Second one needs 12V DC to operate the relays.
Other: This board has Optical Isolation (runs through opto-isolator chips). You need to check if the relays operate with a HIGH(1) signal or a LOW(0) signal. Many of this type of relay board operate on a LOW level so that you can guarantee that no relays will activate at power-on time.
There is some example code for this here:http://arduino-info.wikispaces.com/ArduinoPower
(look for "More Complex Systems)
Let us know how this works out for you...