Does the ADA require that an applicant or employee with a disability be qualified for the position?

Yes. The ADA defines qualified to mean a person who meets legitimate skill, experience, education, or other requirements of an employment position that s/he holds or seeks, and who can perform the essential functions of the position with or without reasonable accommodation. Requiring the ability to perform "essential" functions assures that an individual with a disability will not be considered unqualified simply because of inability to perform marginal or incidental job functions. If the individual is qualified to perform essential job functions except for limitations caused by a disability, the employer must consider whether the individual could perform these functions with a reasonable accommodation. If a written job description has been prepared in advance of advertising or interviewing applicants for a job, this will be considered as evidence, although not conclusive evidence, of the essential functions of the job.

For additional information, take a look at the following resources:

FAQ: What are the  “essential functions” of a job?

FAQ: Who is a “qualified individual”?

FAQ: Can an employer be required to modify, adjust, or make other reasonable accomodations in the way a test is given to an applicant or employee with a disability?