Annuities are tax-deferred, not tax free. They grow without income tax until funds are withdrawn.
In a non-qualified annuity (that means it is not in an IRA or other retirement account) funds withdrawn are taxed on a last in, first out basis unless annuitized. This means all profit comes out first and is taxes before your original money is returned, which is not taxed (it was taxed before it was put in). If you annuitize, then profits and original basis are pro-rated.
Example: You put $20,000 into a non-qualified annuity and it grows to $40,000. If you take money out without annuitizing, then the $20,000 profit comes out first and is taxed at ordinary tax rates before the principle. If you annuitize, half of each payment is taxable profit and half is retrun of principle.
Note: If you take money out prior to age 59 1/2 you will pay a penalty in addition to the income tax unless taken out under Section 72(q), equal and periodic withdrawls.