Question is in the title, can An arduino UNO control a 16 channel relay board?
Yes, if the board accepts logic TTL (5V) levels.
Most probably!
Without knowing enough about the specific relay board (maybe a link to the product page where you bought it/are looking to buy it?), it's hard to tell. Don't be put off by the 14 (0-13) outputs of the Arduino Uno, you can also use all of the analog pins as output too, giving you 20 IO lines you can use.
It may also be helpful to know what else you're trying to do with this; for example if you need a serial connection, then pins 0 and 1 may be unavailable to you, and similarly if you need SPI or I2C communications, or an analog read, or even PWM outputs, then they all impact the pins you can use.
If your relay board is capable of being driven from 5V, then the Arduino shouldn't need any extra components to support it (though that does depend on the current - again without knowing your specific relay board it is hard to tell!), and if it's a different voltage then it should still be possible, and you may need to add some transistors so the voltages play nice.
Hope that helps!
Tom
A 16-relay board can give problems. Two 8-channel might be better.
http://forum.arduino.cc/index.php?topic=321782.0
More pins
http://forum.arduino.cc/index.php?topic=180144.0
Leo..
is the board and I also am using a bluetooth module
That board has a weird circuit diagram. Follow the link in post#3
Wrong design. Optos, but no opto isolation.
Could also be dangerous in some situations, because the relays could all turn on when you turn your Arduino off.
There is a document link just below the youtube windows on this Sainsmart page.
Leo..