A disease is a pathological condition that negatively affects the structure and/or the function of all or part of an organism, and it is not due to any immediate external injury.
Examples of diseases: Huntington disease, heart disease, cancer.
A disorder is a disruption to regular and normal bodily function and structure. The mind and body is in a state of confusion.
Examples of disorders: mental disorder (and almost all the subtypes like anxiety disorders, personality disorders, post-traumatic stress disorder, mood disorders), chromosome disorder.
A syndrome is a collection of signs and symptoms associated with a specific health-related cause. This group of symptoms quite consistently occur together. A syndrome is directly defined by the set of symptoms.
Examples of syndromes: irritable bowel syndrome (IBS), Down Syndrome.
A disability is any condition that makes it more difficult for a person to do certain activities or interact with the world around them. These conditions, or impairments, may be physical, sensory, mental, cognitive, developmental, intellectual, social, or a combination of those.
Examples of disabilities: vision impairment, physical disability, intellectual disability, specific learning disability. (Many diseases and disorders mentioned above can lead to disability)
A dysfunction is an abnormality or impairment in the operation of a specified bodily organ or system, it can also be a deviation from an accepted social behaviour.
Examples of dysfunctions: sexual dysfunction, cognitive dysfunction, kidney dysfunction.
An illness is an unhealthy condition of the body or the mind that lasts for a period of time. It appears to be a broader term that can include most of the above mentioned (?).
Examples of illnesses: flu, diarrhoea.
Notice that the definitions are not clear cut, especially in Psychopathology.