Most first upgrades go wrong in one of three ways: the headphone needs more power than the source can give it, the wrong enclosure for the room, or a pair that measures well and hurts after an hour. This page is about those three things, in that order.
Impedance, and whether your source can drive it
Impedance is how much the headphone resists the current an amplifier pushes into it, measured in ohms. Low impedance headphones, anything from about 16 to 60 ohms, draw more current at a given voltage, which is why they get loud from a phone. High impedance headphones, 250 or 300 ohms, need more voltage to reach the same level, and a small source runs out of voltage before it runs out of will.
Impedance on its own does not tell you the whole story. Sensitivity, usually printed as decibels per milliwatt or per volt, tells you how loud a headphone goes for a given input. A 300 ohm headphone at 105 dB/mW is easier to drive than a 32 ohm one at 90 dB/mW. Read them together.
The practical rule: if a headphone is above about 100 ohms, or below about 95 dB/mW, assume it wants an amplifier. That is not audiophile superstition, it is headroom. An underpowered headphone sounds thin and flat at the top of its volume range because the source is clipping, not because the driver is.
Open back or closed back
An open-back headphone has a grille instead of a sealed cup. The driver works against the room, which generally gives a more spacious, less boxed-in presentation. It also means everyone within a few metres can hear your music, and you can hear them. In a shared office or on a train, an open back is the wrong tool.
A closed back seals against your head. It keeps sound in and some of the room out, at the cost of a smaller sense of space and, in cheaper designs, a boomier low end from the trapped air. If you listen anywhere other than a room with the door shut, start here.
Nobody can tell you how much noise a closed back keeps out without a measured test report, so be careful with any shop that quotes an isolation figure without one.
Comfort beats specification
Three numbers predict whether you will still be wearing a headphone in the third hour: weight, clamp force and pad depth. Anything over about 350 grams starts to be noticeable on a long session. Clamp is rarely published, but pad depth is a good proxy, if your ears touch the driver, they will get hot and sore.
Pads are also the part that changes most over time. Foam compresses, which increases clamp and moves your ear closer to the driver, which changes the sound. A headphone with replaceable pads is a headphone you can reset in three years for the price of a pair of pads.
A short checklist
Work out where you will listen. Check the impedance and sensitivity against what you will plug into. Pick the enclosure that suits the room, not the one that measures best. Then check the weight and whether the pads and cable can be replaced. If a pair passes all four, the rest is preference.