A perpetuity is a type of annuity where there is no end to the payments. It may have fixed or growing payments depending on its nature. For example, a rental property will give you a fixed amount every month. Meanwhile, a government bond will result in an increasing amount after each period as time goes on.
In contrast, an annuity has a limited number of payments over its life span before it ends completely. For example, the end-of-year bonus from your job or pension from your company is an annuity since it has a fixed number of payments.