What the day-of-year number means
Day of the year is the selected date's ordinal position within its calendar year. January 1 is day 1. December 31 is day 365 in a common year and day 366 in a leap year. The calculator also shows the days elapsed before the date and the days remaining after it so the boundary convention is explicit.
How leap years affect the result
A Gregorian leap year is divisible by 4, except century years must also be divisible by 400. February therefore has 29 days in years such as 2024 and 2000, but not 1900 or 2100. Dates after February 28 have an ordinal number one higher in a leap year than in a common year.
ISO weeks, quarters, and year progress
The ISO week uses Monday as the first weekday and assigns week 1 to the week containing the year's first Thursday, so its ISO week-year can differ near New Year's Day. Calendar quarters are Jan-Mar, Apr-Jun, Jul-Sep, and Oct-Dec. Year progress is the ordinal day divided by the number of days in the selected year.