The main issue with how emotional and other psychological disabilities may be covered under the Americans with Disabilities Act has to do with the concept of "reasonable" accommodation. Businesses are required to provide reasonable accommodations which do not place undue burdens on them.
Thus, for example, someone with a mood disorder well-controlled by medication, who might need one a hour a week to meet with a psychiatrist during the work week would be fully covered for nearly all imaginable jobs under ADA; this is not a particularly difficult accommodation.
Someone with mild Asperger's who does not work well with the public could be accomodated in many jobs that do not require regular public interactions (computer programming, manufacturing, construction), but might not be able to be accomodated in a job such a real estate sales.
It really varies a good deal depending on the specific job and specific disability.