The 16 PWM signals means that you can run up to 16 separate servos, but you would need an external 5V power supply with enough current to supply all of them. The Omega by itself can power 1 or 2 small servos under very light load, although it can get a standard size servo to turn a door lock here!
You would definitely need to find the maximum current draw of each servo (usually in their specs), and make sure your external power supply can provide more than the sum of all maximum currents (for safety).
Labelling each wire is very handy, you can either label by part/location or by #; the connectors on the Expansion are marked from S0 to S15.
Full technical details are available in our Servo Expansion Tutorial.
PS: Please post your project when you're done, we'd love to see it!