The first thing to remember is that this kind of model represents an analysis rather than a rule. It has been formed by examining a number of prominent companies (in this case, they picked firms like Lego, Microsoft, Sony, & Xerox) and looking for a common pattern in the way they do things.
(These analyses are interesting in themselves -- the Microsoft article explains the thinking behind Office 2007 and its infamous ribbon bar interface, and demonstrates a staggering arrogance on the part of their designers, aiming for a simpler interface by removing (useful) advanced features rather than just hiding them.)
The design process model is known as the 'double diamond'. The image below is my own adaptation of their diagram, showing the role of the brief more clearly.
Although it's not stated on the diagram, the whole process usually starts with a problem that needs to be solved.
The first phase -- Discovery -- deals with research & brainstorming, including market research and an analysis of user needs. As this stage progresses, the number of options available will expand rapidly, driven from the initial brief.
The second phase -- Definition -- begins with a feasibility review, which will start to filter & discard ideas which cannot be carried forward. This may come down to cost or materials, or time. Anything which deviates from the brief will probably need to go, too. This may include pitching & prototypes, and concludes with a small number of ideas to be offered as a solution to the problem specified in the brief.
Development takes an approved & signed-off idea and starts to go into details. This may lead to a number of prototypes or demo versions, which will need to be tested & reviewed.
Finally, the product is delivered. At this stage it is necessary to reduce options, and select the version which will be delivered to the client or customer. Additionally, this will often require the setting up of a feedback system to help evaluate the effectiveness of the solution in its real-world use.
This is not the only model to describe a design process, but it dovetails nicely into the spiral model (previously posted here) which describes the rapid application development iterative design process, common in computer programming.
Here, the whole double-diamond process will be repeated -- in differing levels of detail -- in each full rotation around the centre.
These models are useful, because they present a framework when planning a project, and highlight the importance of milestones (e.g. feasibility review) on.