Many schools for the gifted offer enrichment in academic subjects. Others specialize in the arts. Most of the students in these schools excel at their school work and in their artistic endeavors. They thrive in a school setting where they don't have to waste time on non-essential courses. The extra time gained is spent on music lessons, rehearsals and studio time.
Several cities offer 'street schools' modeled after the highly successful Denver Street School. The Cristo del Rey model sponsored by the Jesuits offers another option for 'at risk' teens.
Special needs schools cover a wide range of learning issues. Everything from dyslexia to ADHD is dealt with by highly trained professionals. Schools such as The Forman School in Litchfield, Connecticut, are famous for their great results.
Be warned: alternative schools are not inexpensive. The one on one teaching which most of these schools use escalates the cost of educating a child. Is it worth it? Ask the satisfied graduates and their parents.
While it is extremely difficult to define exactly what one means by non-traditional schools, Montessori, Waldorf, Carden and schools following a progressive approach are generally thought of as non-traditional schools. A discussion of this genre of private school can be found here.