An annuity is qualified if it is part of a tax-advantaged retirement plan. This includes 403(b) plans for employees of government and non-profit agencies, defined-benefit pension plans or individual retirement accounts. In most cases, your employer deducts these contributions from your pay, and the income taxes on them are deferred until you take the money out. There are some situations, such as in the case of a self-employed minister, where someone might make after-tax contributions to a 403(b) plan. In that case the contributions would be deductible on taxes.
Non-qualified annuities are the typical types of private annuities that people buy on their own to provide an income-stream in retirement. For example, you might pay premiums over several years to build up an annuity account to pay you monthly income once you retire. Or, you could take a portion of your retirement nest egg, invest it in an annuity in one lump sum, and immediately start receiving payments. Even though you buy these products with after-tax dollars, the Internal Revenue Service does not allow you to take a tax deduction for the premiums you pay.