Depending on what test equipment you have available, before you apply any power, you can test it's resistance, inductance or capacitance (probably the first 2 are most useful in this case since it looks to be a coil of some sort)
An LCR meter (Inductance Capacitance Resistance) is a great tool for testing unknown components such as this. I have a Mastech MS5308 (can be found for £133 on eBay) and can highly recommend it. For the price it is comparable with more expensive meters (from e.g. Agilent, BK Precision, etc) comes with proper 4 wire kelvin clips, 4 wire SMD tweezers, an isolated USB link and case.
It tests at frequencies of 100Hz, 120Hz, 1kHZ, 10kHz, and 100kHz (the more test frequencies the better - many lower quality meters only have a couple and do not test at 100kHz, which is the frequency most capacitors are tested at for the datasheet specs)
It has a minimum resolution of 0.01pF, 1nH and 1mΩ which is very impressive indeed, tests ESR (Equivalent Series Resistance), Q, DF, θ and has a sorting/tolerance mode you can use to quickly find parts that are out of rated tolerance. For comparison the Agilent U1731C only tests at 120Hz and 1kHz, and has a minimum resolution of 0.1pF and is over £200. Anyway, you might guess I'm extremely happy with mine, but have a browse around before deciding just in case I'm talking rubbish :-) )
Also, you could carefully peel a little of the covering away to reveal the windings (assuming it is a coil)
When you do apply power, preferably use a current limited bench supply and ramp up slowly. Try swapping leads around to see if this makes any difference (just in case there's a series diode hidden in there somewhere)
A function generator may also be of use, possibly combined with an amplifier in case it requires AC drive.
Having said all the above, it does look like either an electromagnet or part of a solenoid/relay assembly (but it could be a buzzer, or...)