The most basic and among the most useful type of annuity is the lifetime income annuity. In it's purest form, you pay a lump sum premium to the insurance company, and they guarantee you a specific annual or monthly income every month or every year, for as long as you live. If you die sooner than expected, the annuity company generally pays any premium left over to your heirs. If you live longer than expected, the annuity company must continue paying your income out of the general fund. Your income becomes a general obligation, guaranteed by the general fund of the insurance company. Your own investments in stocks, bonds and mutual funds can be spent down to nothing, but with a life income annuity, the insurance company must keep up the payments to you regardless of market conditions - even if it loses all your money in its own investments. Remember, an annuity is a contract - not an investment.
A deferred annuity is simply a contract obligating the insurance company to provide income at some future date that you may select. Any money you have contributed to the annuity can compound according to the terms in the contract, tax-deferred, until it's time to "annuitize," or convert your paid in and/or credited premium to income. Deferred annuities let you contribute premium over time, and are popular retirement savings vehicles. With an immediate annuity, the income, of course, starts right away, as soon as you have contributed.
A 'fixed' annuity earns a minimum guaranteed rate of return. A variable annuity can gain or lose money based on the performance of the 'subaccounts' you select. These can be in treasuries, money markets, bonds, the S&P 500, or tied to some other major index that you select. Variable annuities allow you to access the guarantees and tax advantages of annuities while still potentially participating in the upside of stocks and bond- type investments over time. One 'hybrid' is the fixed index annuity, which provides a guaranteed 'floor' for your returns over time, but also credits you a portion of any positive market returns in excess of that floor. These FIAs can be complex, so understand what you're buying.
Annuities provide some certainty of income in retirement (though different kinds of contracts are available with varying degrees of income fluctuation or risk). They also provide the leverage of tax deferral while the assets in the annuity compound. Contributions are taxed as ordinary income, but you aren't taxed on the portion of income attributed to return of premium. You are only taxed on a portion of the income attributed to growth. Annuities may also provide substantial protection from creditors, depending on your state.