If you're listening to analogue recordings like cassette or radio, you probably won't need to worry too much about frequency response. For digital recordings and home theatre, however, look for something that claims to be at least 20Hz to 20,000Hz. Many headphones also provide an equalisation setting that is designed to simulate a listening environment, and there are two common algorithms. Free field equalisation (FF) is intended to simulate an open space without echoes or reflections, and the sound source positioned directly in front of the listener. The other type of equalisation is called diffuse field (DF) which is designed to reproduce the listening environment of a room with reflecting walls, and is generally more natural sounding than FF equalisation. Although DF is an industry standard algorithm for sound reproduction, it does not suit all recordings or listeners, so it is best to compare a few types before making a decision to buy headphones that incorporate it.
If you're after high quality headphones for wearing on the move, you may want to choose smaller cups which sit on the ear, although a lightweight "closed-back" model is still an option. One downside with open style headphones is that they are subject to bass leakage, which is why some manufacturers build bass boost into either playback devices or the headphones themselves. If you want something that cuts out external sounds but is still compact, an alternative to sealed cups is to use noise-cancellation headphones instead. Noise-cancellation headphones are designed for use in places with significant ambient noise (such as aeroplanes), and they work by using miniature microphones that record the ambient noise and mix a cancellation signal inside the cups. These are generally very lightweight and compact, but are often fragile as well. So if you are planning to buy a lightweight model, make sure it comes with a protective bag or case.
Another option is a pair of in-ear monitors (IEMs). These are similar to ear buds, resting inside your ear rather than on top of it; however, they go one step further and actually sit inside your ear canal. This can be a little uncomfortable for first time users, but once you get the hang of them, IEMs provide incredible sound quality and isolate you very well from external noise.
You won't need a long cable if you're carrying a portable media player around, but you might need a mini-jack adapter on the plug. Some headphones have screw-on ones which make them harder to lose. If you are looking for the best possible sound from a portable player, however, you might want to consider a portable amplifier. These are very compact and provide the level boost you need to get the most out of your headphones. On the down side, they are likely to cost at least as much as the headphones themselves as they are usually designed for professional use.