A perpetuity is an annuity in which the periodic payments begin on a fixed date and continue indefinitely. Fixed coupon payments on permanently invested (irredeemable) sums of money are prime examples of perpetuities. Scholarships paid perpetually from an endowment fit the definition of perpetuity.
The value of the perpetuity is finite because receipts that are anticipated far in the future have extremely low present value (today's value of the future cash flows).
For example, UK government bonds that are undated and irredeemable (e.g. War Loan) pay fixed coupons (interest payments) and trade actively in the bond market. Very long dated bonds have financial characteristics that can appeal to some investors and in some circumstances, e.g. long-dated bonds have prices that change rapidly (either up or down) when yields change (fall or rise) in the financial markets.
A more current example is the convention used in real estate finance for valuing real estate with a Cap_Rate. Using a cap rate, the value of a particular real estate asset is either the Net_Income or the Net_Cash_Flow of the property, divided by the cap rate. Effectively, the use of a cap rate to value a piece of real estate assumes that the current income from the property continues in perpetuity.
See also: Present_value