"Mixed dementia" simply means that a person has more than one disease mechanism at work.
"Dementia" is really a name of symptoms, it is not actually a disease in itself - it actually translates as "de-mentia - without mind". Confusingly, some disease that cause dementia has dementia in their names and this is often because the name simply means "dementia of cause xxxx"
It is possible for a person to have more than one of these diseases, since having one form does not preclude developing another.
By far the most common diseases are Alzheimer's Disease and Vascular Dementia (this is simply dementia causes by problems with blood supply to the brain, usually, the small vessels).
Therefore, most people with mixed dementia have Alzheimer's and Vascular. The other causes of dementia are much rarer.
Both of these diseases are usually diagnosed simply by looking at the patient's symptoms and, importantly, how those symptoms progress over time. There is no definitive test - things like blood tests are simply used to rule out diseases that can mimic such symptoms (such as a vitamin deficiency), because these are treatable. Brain scans can sometimes help to diagnose vascular disease.
In the early stages both of these diseases cause very similar symptoms and it can be very difficult to distinguish between the two. However, they normally progress in different ways: Alzheimer's as a steady and often slow decline; vascular by sudden declines, which can be severe, with periods of relative stability in between. These declines tend to be unpredictable.
Thus if someone is showing a progression pattern that shows features of both diseases, they are presumed to have "mixed" dementia.
The other diseases are, as I said, much rarer and tend to have distinctive symptoms and patterns of progression that mark them out - but it is still possible to get, say, Lewy Body disease and Alzheimer's.
The risk of both Vascular and Alzheimers rises dramatically with age; Alzheimer's is very rare (though not impossible) for people under 60 and most people tend to be over 80. It is predominantly a disease of the elderly.