There are two forms: one that pays once at maturity (the zero-coupon form), and one that lines up one-year options for each year. In the zero-coupon form, at maturity the buyer receives the notional amount times the part by which inflation over the whole term exceeds the strike compounded over the number of years. In the annual form, the buyer receives each year the notional amount times the part by which that year's inflation exceeds the strike.
Consider a hypothetical example with a notional amount of 10 billion yen, a strike of 3% a year, and a term of one year. The payoff is 200 million yen if inflation for the year is 5%, and 0 yen if it is 3% or lower. The buyer pays a premium at the start, so if inflation does not exceed the strike, the loss is the premium alone.
An inflation cap refers to a rate of inflation calculated from a price index, and the payoff is determined by the index from 2 to 3 months before maturity.