Low-impedance (under 50 ohm) headphones will play well at home with your receiver and on-the-go with your smart phone, and that's why the vast majority of in-, on-, and over-the-ear headphones are low-impedance designs. High-impedance models are much better suited to home than on-the-go use. Sennheiser and Beyerdynamic are two companies that offer high-impedance consumer headphones with up to 600 ohm-rated designs.
I used my solid-state NAD C 316BEE integrated stereo amplifier for most of my high-impedance headphone listening tests. I also noted that tube headphone amplifiers have a special affinity for high-impedance headphones; my budget-priced Bottlehead Crack headphone amp sounds astonishingly good with my 300-ohm Sennheiser HD 580, and 150-ohm . My 600-ohm Beyerdynamic T-1 headphones' sound had extraordinary dimensionality with the Crack, and resolution of fine detail was terrific.
Beyerdynamic is the only headphone company I know that offers models, like the DT880 and DT990 Editions for example, in three different impedances: 32-, 250-, and 600-ohm versions. Each version, regardless of impedance, is the same price. So sure, if you will use the headphones with portable battery-powered devices and at home with a receiver or amplifier, get the 32-ohm version. But if you'll only listen to them at home, go for the high-impedance 600-ohm versions, because to my ears the high-impedance DT 880 Edition and DT990 Edition headphones sound better than the 32-ohm versions. The high-impedance versions sound more transparent and clearer, bass definition is better, and the soundstage is more spacious.
The impedance of a headphone is largely determined by thevoice coil, and for Beyerdynamic's high-impedance models the voice coil's wire is super-thin, just 0.018mm, half the thickness of a human hair. Beyerdynamic's Senior Product Manager Gunter Weidemann explained that manufacturing voice coils from fine wire is extremely complicated, so the company makes all of its high-impedance headphones in Germany.
The thinner wires have more windings (layers of wire) on the voice-coil than the lower-impedance Beyerdynamic headphones, which have thicker and heavier, easier-to-manufacture voice coils. The lower moving mass of the 250- and 600-ohm headphones' voice coils is lighter than the 32-ohm models, and the lower mass is part of the reason high-impedance headphones sound better. The smaller diameter of the 600-ohm voice coil wires allows the wires to fit tighter, so there's less air between the windings, and that makes the electromagnetic field of the voice coil stronger. All of that reduces distortion for the high-impedance versions compared with the low-impedance headphones.
Weidemann pointed out that while most speakers' impedance range covers a narrow band between 4 and 8 ohms (or 16 ohms on my Zu Druid V speakers); headphones can be as low as 8 ohms and as high as 600 ohms! That's why Beyerdynamic's high-end headphone ampshave a switch to optimize their sound for use with low- or high-impedance headphones.
Weidemann's explanation for why battery-powered devices perform best with low-impedance headphones is based on this: Most portable players use from 3-volt to a maximum of 5-volt batteries, that means the signal voltage at the output is low, but they can still deliver high current with low-impedance headphones, and the sound will be fine. However, if you plug in a high-impedance headphone to a portable device the voltage and current will be low, so the headphone won't play loudly at all, and dynamics will be reduced.
Summing up, if you're interested in headphones that'll be used with portable (battery-powered) music players and at home with a receiver or integrated amp, stick with low impedance, under 50-ohm headphones. If it's going to be a strictly stay-at-home headphone, high-impedance models from Beyerdynamic, Sennheiser and other brands should be considered.
It's worth noting that Beyerdynamic and Sennheiser's very best headphones are only available in high-impedance versions. The Beyerdynamic T-1 is a 600-ohm headphone, the Sennheiser HD-800 is a 300-ohm headphone; no lower impedance versions are offered.