Most likely you already have headphones at home or for general office use. However, to suit the needs for audio recording use, there are some important factors you should consider.
1) More Even (flat) Frequency Response
Usually, headphones commonly used for listening music are preset with EQ, biasing to certain frequencies. However, for audio recording use, flat frequency response, usually works better. A flatter frequency response headphone simply gives you a more accurate picture of what's being recorded from the mic.
Here's a chart showing how different models of headphones can be biased on different frequencies. (figures from "HeadRoom" webiste)
What we're looking for is the Reference or Monitor Headphone type, commonly used in many audio studios, like this one shown below by Sony. Obviously, you see the frequency response is much flatter, especially on the vocal range from 50Hz up to 4KHz to 5KHz; you notice the dBr values are +/- less than 5.
2) Full Closed Headphones
Very often, the person doing the recording with a mic will also at the same time listening to something - either music, his own voice or others' voices. That means, he needs a headphone.
Full closed type headphone is made to avoid bleeding out any sound, so that it will not be picked up by the mic. Some monitor headphones are called "Half-closed", which means not totally cut the sound bleeding. Sometimes, they could be acceptable and more cost effective. However, most personal use headphones do not have such ability, and so will not be suitable for audio recording use. This is especially true when you're using a condenser mic, which is very sensitive to any sound, even if it is just the leaking sound from your headphone.
3) Durability and Replacement Parts
Studio use headphones are made for heavy duty, last for years. Consumable accessory items like ear pads and so forth are available and can be easily replaced. Such features are lacking in home entertainment type headphones.