Annual paid leave. Not to be confused with summer leave. Each incoming JET starts off with 20 days of nenkyuu. Nenkyuu can be taken in full-day, half-day (3.5 hour), 2-hour, or 1-hour increments. Nenkyuu gets rolled over between each year of appointment, and 20 days get added at the beginning of each year. Maximum number of days is 40.
Person A is a 1st-year whose appointment is from August 3, 2015 to August 2, 2016. She used up a bunch of nenkyuu this year and has 5 days remaining as of August 1, 2016. On August 2, 2016, Person A has 25 days.
Going into her 2nd year, Person A wants to save up her nenkyuu. If she does not use up any of her leave, she will still have 25 days remaining at the end of this year of appointment. At the beginning of her 3rd year, she will receive 20 more days. Her new total would be 45 days, but that total would be rounded back down to 40.
For a more detailed explanation, see: What is nenkyuu?