A qualified plan is simply one that is described in Section 401(a) of the Tax Code. The most common types of qualified plans are profit-sharing plans (including 401(k) plans), defined benefit plans, and money purchase pension plans. In general, your contributions are not taxed until you withdraw money from the plan.
In a non–qualified plan, there are no deductions, but the principal is never taxed twice. Instead, the interest is taxed once withdrawn.