My mother has alzheimer's. How do I explain this to my children?

It depends. Explaining a serious illness to children really depends on the age and maturity of the children. A start is to explain that grandma's memory isn't what it used to be so we need to be more patient with her. Other considerations include the stage of your mother's illness. In general, encouraging children to be kind and patient with older adults is a good lesson for all.
Use a taperecorder. Tell them that our memory is like a tape recorder (or other recording machine if they do not know tape recorder any more). In alzheimer's disease the tape recorder is broken and, therefore, people do not remember recent events. However, they can still remember the past when the tape recorder was working.