The season of summer begins with the summer solstice. The summer solstice is the day in which the most amount of sunlight occurs during the day. In the Northern Hemisphere, the summer solstice is around June 21st. In the Southern Hemisphere, it takes place around December 21st.
Summer solstice takes place when the Earth's axis is angled the most towards the Sun. This is why the seasons are different in each hemisphere, due to the natural planetary rotation.
Each season, summer, winter, autumn and spring begin with either an equinox or solstice. The dates in which these occur vary slightly due to the shifting of the calendar from year to year.