In BDSM play, a dungeon is any space set aside for scene activities. It may or may not be underground and may or may not resemble a stereotypical dungeon from a historical film - it could, for example, be a spare bedroom with bedroom-style decorating.
The image shown is from The Edge, a dungeon for hire in Devon 
Dungeons at clubs
Almost every BDSM or fetish club will have a dungeon play space, depending on if the club is more play or dance orientated the quality and amount of equipment will vary greatly as will the type of play taking place.
Dungeons for hire
There are a number of purpose built dungeons for hire in the UK. Usually they can be booked either during the day or overnight. The cost varies from establishment to establishment but is typically between £200 - £300 a session. Prices are expensive but comparable to the cost of hiring a suite at a hotel for the night. Dungeons vary greatly in what equipment they have and how much space there is to play. It is well worth making sure that the dungeon you are thinking of hiring will suit the sort of play that you like doing.
Dungeons at home
The vast majority of people do not have the space or desire to have a purpose-built dungeon in their home. It is however possible to have rooms that can be vanilla 95% of the time and then be adapted for use as a dungeon when the need arises. For people that play at home having a space that can be used as a dungeon can greatly enhance the intensity and enjoyment of their play.
What an individual wants from a dungeon will very much depend on what aspects of BDSM are important to them. However, in all almost every case, the right atmosphere and enough space to play will be one of the first things to plan. This can be acheived simply by rearranging the furnature in the room and the sensible use of light and candles to create a mood. Planning how you will use the space before you start playing will allow you to concentrate more on scene itself and less on the mechanics of what you are doing. Getting the sub to set up the play space beforehand is also a great way to build the sense of sexual anticipation in the sub and to prepare them emotionally for the session.
Specialised BDSM furniture can easily be added to a home play space and can make a massive difference to the intensity of a scene. Most types of BDSM furniture can be bought in a form that can be easily dismantled for discreet storage. The most common items of furniture are Spanking benches and St Andrews crosses. It is also possible to buy furniture that can be used in dual mode: for most of the time, it looks like a normal piece of furniture, but it can be converted to somthing more kinky when required.
If you are into bondage then having an overhead fixing point is a real bonus. This can be acheived in a number of ways; The simplest method is to use door frames, these can be used for attaching door hangers from which a person can be tied to (although not suitable for suspension bondage). A more versitile option is to use scaffolding which can be used to build a frame on which a person can be bound or suspended from (this can then be dismantled at the end of the scene for storage). Permanent fixing points can be created using expansion bolts drilled into the walls, and from which chains and ropes can be attached. However, this will only work with brick walls or similar. Once fitted and painted they are inconspicous and can also be hidden behind pictures if you are worried about awkward questions being asked. Finally if you can locate the position of your ceiling joists then ring bolts can be screwed into them for overhead suspension points. If you are considering this sort of fixing point then considerable care needs to be taken with how you go about doing it. Should a suspension point fail during a session the person being suspended could suffer serious injuries as a result. Fetters sell a joist hanger designed to safely attach a suspension point to a joist.
If you are into more traditional rope based bondage (hogties etc.) or you want your sub to spend any amount of time on their knees, having a large padded mat is a useful addition. Ideally this should be waterproof and be wipe cleanable. Floors, even carpeted ones, are usually too hard to leave someone bound on for any length of time; this also applies to having someone kneel for any length of time. A padded mat will provide that extra level of support to prevent pressure problems at the joints and other points of contact. Finally a waterproof mat allows you to use oils and wax without having to worry too much about runining the furnature or carpet.
Technology can also be employed in designing a home play space. Using software and hardware designed for survailence and home automation anyone with basic computer and electrical skills can apply them to BDSM. Most compoments required can be sourced from ebay for a few tens of pounds. What you do with the technology will of course depend on what aspects of BDSM interest you the most, they are likely to appeal to people into humiliation and D/s games.
For about £100 it is possible to buy a low cost video server and camera. This will allow you to monitor your slave from another room. The addition of a basic home automation system will allow you to turn on and off a TENS unit or vibrator thus punishing him/her by remote control. If you have a broadband connection with a static I/P address you can broadcast the humilation of your slave to a wider audience. There are also a number of sex toys specifically designed for this sort of thing, see Teledildonics.