What is a profound intellectual and multiple disability?

Finding a definition of profound intellectual disability is by no means easy. In some places, people with PID are people who have an IQ of 20 and under, however, establishing that somebody has this level of intellectual functioning is very difficult. So essentially this definition is not terribly functional in practice because there are few tests that can discriminate this level of ability, in particular when you also throw in physical and sensory impairments as well. Some researchers have chosen an cognitive age equivalency to determine if somebody has a profound ID. These cognitive age equivalencies have varied across researchers and nations with some people reporting it as under 12mths, 24mths, or IDC-10 reporting 3 year equivalence (I am aware of some of the pros and cons of using age equivalency but shant go into it now).
The definition that I find best at the moment is the one by PMLD Network in the UK (in the UK an intellectual disability is called a learning disability):

Children and adults with profound and multiple learning disabilities have more than one disability, the most significant of which is a profound learning disability. All people who have profound and multiple learning disabilities will have great difficulty communicating. Many people will have additional sensory or physical disabilities, complex health needs or mental health difficulties. The combination of these needs and/or the lack of the right support may also affect behaviour.

This definition, though giving a clearer idea of the population, still does not say what it means to have a profound intellectual disability. Having a profound ID means that it is difficult for the person to learn new things. It means it is extremely difficult for them to concentrate, plan, problem solve, and remember. It does not mean that a person can not do these things, but it may take much much longer, and the person may need very different teaching techniques. Sometimes the focus for people with PIMD may not be on trying to teach them a new skill, but on how the people around them support them to do things in a different way (i.e. compensating for the skill deficit). It means that people around the person with a PIMD need to think about different ways that they will share meaning with person. Meaning may not be carried through words, instead meaning may come through actions and experiences.

The "multiple" part of the definition means that the person, in addition to their cognitive impairments, may have physical, sensory, and other health issues. Researchers have reported that the more severe an intellectual disability a person has, the more likely they are to have hearing, vision, physical and health issues such as epilepsy or gastrointestinal issues.

However, despite the cognitive and other difficulties that a person with PIMD has, they are still a person, a family member, a consumer, a friend, and a valuable human being.