Capgras syndrome (CS) is a remarkable psychiatric syndrome first reported in schizophrenia in the 1920s. It has also been called the "delusion of doubles"; the patient becomes convinced that someone close to him/her (usually spouse or close family) has been replaced by an almost identical-appearing imposter (if you've seen Invasion of the Body Snatchers, the 1956 sci-fi film, this will sound familar). Although first reported in schizophrenia, CS is almost certainly much more common in Alzheimer's disease (AD), with prevalence probably between 6 and 15%, usually in the moderate stages of the illness. In AD patients, the delusion of replacement for malevolent reasons is often absent ("this is Betty, she looks a lot like my wife, Elizabeth, but she's a different person") and may involve familiar places as well as people ("this isn't my house, take me home"). The more general term is "reduplicative paramnesia"; CS is reduplicative paramnesia for people, with a delusional interpretation that something fishy is going on.
The pathophysiology of CS isn't fully understood, but is likely related to a dissociation between more cognitive and memory-based visual recognition (like you'd recognize a map of Florida) and emotion-based recognition (like the feelings that seeing your child would engender). The latter probably includes what we perceive as familiarity. So, the theory is that if a person matches the visual image of your spouse but doesn't look familiar or elicit an emotional reaction, someting is funny, and maybe he/she isn't your spouse at all. Both recognition systems reside in the right posterior cortex, which is quite impaired in AD.
There's no specific treatment for CS, or for reduplicative paramnesia, it doesn't tend to respond to antipsychotics or other medications. It is probably suggestive of a diagnosis of AD, and it's good to be able to explain what's going on to the family.