What are the differences between an AC and DC coil relay?
The DC coil relay has a resistance from the copper wire typically used. The current is limited by that resistance. 

AC coil relays have inductance as well (the DC relays have inductance as well, of course, but it does not affect the 'on' current). They also typically have a shading ring that acts as a shorted turn in order to cause a magnetic field 90° out of phase with that from the coil, so that the total magnitude of flux does not drop to zero at the zero crossings. 



Edit: As Andy says, an AC relay will work on (greatly reduced) DC. Of course you can also make a DC relay work on AC by adding stuff to it (if you use a capacitor filter you might need only 18VAC (RMS) to operate a 24V relay, and 24VAC would cause it to overheat and fail early).