"The Design Specification sets out in detail the requirements to be met in order to achieve a successful product."

That is it in a nut-shell. Writing one that contains all the facts relating to the product, which takes into account the manufacturer and the user, is more complex. You should try to avoid "leading" the design and predicting the outcome even if, at this stage, the end product is crystal clear in your mind. In fact you are at a disadvantage if this is the case as it will hamper your scope for initial ideas. Equally your specification should contain realistic constraints based on things like the law, the facilities available, time etc.
A good specification, one that you refer to throughout the whole design process, is key to a successful outcome. It is the main control for the whole of your design activity but that doesn't mean it can't be changed. It should be seen as something that may have to evolve if the situation changes. Your starting point must be research; analysing what is being done at the moment or what has been tried in the past, looking at statutory legal requirements, conducting market research through interviews or questionnaires with the intended user.

The following are categories, with descriptions, for you to use to draw up your own specification. They are in no particular order but it seems only right that we begin with performance or function.

Function
What do you want the object to do? How well do you want it to do it? How often? How fast? Is the performance attainable in an economic manner? Try to specify a level that is obtainable rather than the ultimate.

Environment
Consider all aspects of the product's environment (we're talking about "surroundings" environment not "pollution" environment). ie. temperature range, humidity, shock loading, dust, corrosion from fluids, chemicals, noise, vibration, users.

Product Service Life
Are you expecting the product to perform its task 24 hours a day, 52 week a year for five years or once in a lifetime?

Maintenance
Is regular maintenance available or desirable? Will maintenance-free operation prove too expensive to incorporate in the first place? If maintenance is required occasionally will the user be able to gain access to the part and will they need special tools and equipment?

Target Cost
Target costs should be established from the outset and checked against existing or similar products. As one-off manufacturers your costs are going to be very different to everyday manufacturers but that doesn't mean that you're unable to see how costs in real-life might be reduced by using different materials or processes.

Competition
A thorough analysis must be carried out of products that are in direct and similar competitive markets. There is little to be gained from making something which you can easily buy. If your specification shows serious mismatches or deficiencies when compared with what already exists, it might be wise to reconsider.

Quantity
Clearly the scale of production has a massive influence upon the manufacturing processes used. You are all making one-offs but you need to consider how your idea might be adapted for mass-production.

Size
Obvious really, but there are likely to be size constraints for your projects which must be adhered to.

Weight
As above. Particularly important for anything portable.

Aesthetics/Finish
Whatever the product, the customer sees it first, the performance comes later. Aesthetics are difficult to specify in a meaningful way at this stage. You want it to look good, of course, who doesn't? It might be important that it stands out or is visually striking or perhaps discrete, so try to specify colour(s), shape, form, texture of finish.

Materials
The choice of materials should be influenced by the design rather than the other way round. It may be the case that certain properties particular to certain materials are desirable. Conversely there may be strong reasons for not using a material ie. toxicity.

Standards
Does the product need to meet British/International standards? If so copies must be obtained from BSI or ISO to see what needs to be done in order to meet them.

Ergonomics
All products have, to some extent, a man/machine interface. It is therefore important to take into account ergonomic considerations. What height, reach, forces, postures, lighting levels should be considered?

Customer
It is essential to obtain first-hand information on client likes/dislikes, preferences and prejudices. Their input will vary depending upon whether there are already like products on the market or whether the product is breaking new ground.

Quality and Reliability
Both of these are difficult to obtain objective information for prior to manufacture although statistical data for some components, particularly electronic, is available. You can certainly draw attention to those aspects of your project for which quality and reliability are important. A project is only ever as reliable as its weakest component.

Testing
It is important to come up with some criteria that you will use to measure the performance of the product. How can you judge whether your project has met the quality and durability criteria that you have laid down?

Safety
Much of this links in with standards and regulations, but you should additionally consider things like operating instructions and warning labels. You ought to consider what the implications are if your product is misused either inadvertently or deliberately.

That covers just about all aspects of the specification that you will need to worry about. Each point can be considered from the point of view of the user and the manufacturer (you or a main-stream manufacturer if this is a prototype which is worthy of mass production).
You should present the specification as a predominantly written document but use illustrations and diagrams where appropriate.