THE LONGEST DAY ON EARTH
Due to gravitational effect of the Moon, the energy of Earth's rotation is slowly being absorbed by the Moon. Hence Earths orbit is slowing down approximately by 2/100ths of a second per century, Thus making each and every new day the longest day on Earth.
If you want to refer to the longest day as the day with the longest hours of sunlight, then it would be around June 21st, when the sun reaches it's most northerly point. This day is known as the summer solstice. The exact length of daylight hours vary on your latitude, the farther north you go, the longer the day. All this is true only in the Northern Hemisphere, in the Southern Hemisphere all this happens around December 21st.
At the North Pole the sun may not set for a couple of weeks making for one long day, but at the South Pole it's one long night.
Then there is late October where many place set their clocks back 1 hour making for about a 25 hour day.