Very hot days during July and August.
The ancient Romans noticed that the hottest days of the year, i.e. in late July and early August, coincided with the appearance of Sirius - the Dog Star, in the same part of the sky as the Sun. Sirius is the largest and brightest star in the Canis Major constellation, in fact it is the brightest star in the sky.
The ancients believed that the star contributed to the heat of the day. The adjective Canicular means 'pertaining to Sirius', so Dog-days are also called Canicular days. This is first referred to in English in John de Trevisa's work Bartholomeus De proprietatibus rerum, 1398:
"In the mydle of the monthe Iulius the Canicular dayes begyn."