Generally, given the same sensitivity (which may vary, of course), cans with high impedance want to see more output voltage while ones with low impedance can live with less voltage but need more current for the same volume. Thus high-impedance cans will be better suited for home use where sufficiently high voltages are not a problem (generally +/-12V tends to be present) and opamps may be used for amplification, while low-impedance cans will suit portable devices with their lower output voltages and power amp chips (i.e. generally sufficiently high currents).
Critical cases are insensitive high-impedance cans (typically described as "hard to drive") and very low-impedance cans that require insane amounts of current, which at the power buffering possible in small portable devices will lead to higher battery drain. Low-impedance cans also get harder to drive for devices with non-negligible output impedances (typically, home speaker amps and receivers, but also radios that have a speaker of their own) as the voltage drop across the internal resistance rises. The sweet spot is at load impedance = output impedance. (If output impedance --> 0, then only the maximum current will limit the possible output power.) That's why I can wring higher volume out of my HD590 (120 ohm) than either a DT231 (32 ohm) or a HD424 (2k ohm) on an amp with a Z_out of ~140 ohm. (Given a source with very limited output current, however, like a SB AWE64 line out, the HD424 is at an advantage since you can get undistorted playback when the HD590 already is too much for the opamps used. At the same time, it isn't going to be much fun out of a pocket radio running off a single 1.5 V cell - a HD650 wouldn't either -, where the DT231 would be best suited.)
Related issues are the influence of output impedance on frequency response (the impedance of headphones frequently varies considerably with frequency while the output impedance tends to be fairly constant, thus frequency response will be altered as you go to higher output impedances) and the bass roll-off encountered when there are coupling caps in the output and you connect low-impedance cans (the output+load impedances together with the coupling caps form a nice highpass which scales with 1/(R*C) frequency wise, i.e. you'd need to compensate lower impedances with fatter coupling caps), as frequently to be observed with onboard sound and such.