There have been multiple studies comparing the effectiveness of manual brushes and electric brushes.
Although not all electric brushes are the same, these studies conclude that, in general, electric brushes are more efficient in controlling plaque than manual brushes. Theoretically, you can do a very good brushing with a regular hand brush, but the movements of an electric brush make the task easier and more effective.
Also, some electric brushes (Sonicare) produce sonic vibrations that are difficult to mimic with a hand brush! Other electric brushes like Oral-B and Rotadent have small heads that help you access hard-to-reach areas of your mouth. This aspect is more important when you are talking about someone with orthodontic braces or a history of gum disease.