Summer in the northern hemisphere starts on June 21-22 and goes to September 21-22. It is the time of Earth's orbit when the norther hemisphere is tilted towards the sun, enabling that hemisphere to receive more direct radiation than at other times of the year. Direct radiation transfers directly to hotter temperatures; that is why locations around the equator are warmer year-round than area at the poles, they receive more direct radiation.
As to the hottest day of the year, there are shifting weather conditions which influence the day-to-day temperatures, so there is not an "exact" day on which the temperature is guaranteed to be the hottest. Temperature ranges within the above specified seasonal time (summer) would be the closest correct answer that could be given.