A variable annuity is a contract between you and an insurance company. You buy into the variable annuity either in a lump sum or over time, and in return you receive payments over time from the insurance company. The amount of the payments depends on the investment results you achieve by investing in mutual funds offered and managed by the annuity.
Variable annuities are usually sold to retirees who want a fairly regular stream of income. They can be complex arrangements that provide insurance and mutual fund companies with large profits at your expense. Unless you're approaching retirement age, you're likely to fare better with other investment choices.