Alois Alzheimer was a German psychiatrist who was the first to study the neurodegenerative disease, presenile dementia, which was later called Alzheimer's disease. Symptoms of Alzheimer's include memory loss, confusion and the impairment of language capabilities.
The correct pronunciation of Alois Alzheimer in German can be transcribed as Ahl-ois Ahltz-eye-muh.
The "A" at the beginning of Alzheimer is an open "ah" sound while the "R" at the end of Alzheimer is silent because in that position in German it is often not pronounced. A closer approximation of that sound is similar to the "Ai" sound in the word "air", minus the "R", which is not heard.
When adding the possessive apostrophe and "s" to the end, as in Alzheimer's disease, the pronunciation of Alzheimer remains the same, but the possessive is strictly English and is pronounced as you normally would.