There is a widespread custom to study one chapter of Pirkei Avos each Shabbas afternoon during the summer months. (Rema O.C. 292:2)
Some observe the custom for the first six weeks following Pesach, thereby completing the entire tractate of Pirkei Avos once as preparation for Shavuos and "Kabbalas Hatorah." Pirkei Avos contains six chapters, and there are six Shabbasos between Pesach and Shavuos.
In most communities, however, the Pirkei Avos study continues beyond Shavuos and goes to Rosh Hashanah. The custom of continuing the cycle of Pirkei Avos until Rosh Hashana, thereby completing it four times, is intended to recall that Moshe Rabbeinu taught the Jewish people the Torah four times.
Most people study Pirkei Avos afternoons on Shabbas following mincha, while others do so before.
There are several reasons why the practice of studying Pirkei Avos was instituted.
Among the more popular explanations is that the summer months are a more recreational and coolheaded time of the year, making the moral teachings of Pirkei Avos especially appropriate.
It was established to provide the unlearned with an organized structure for Torah study on Shabbas afternoons.
After the Jews left Egypt, they embarked on a period of self-refinement and character improvement. This was critical so they could receive the Torah on Shavuot. While counting the Omer, we, too, try to improve our character. To achieve this goal, we study Avos, the tractate dedicated to piety, humility, kindness, and ethics.
Abudraham says: The custom to read Pirkei Avos and the perek of acquiring the Torah on Shabbos in the weeks between Pesach and Shavuos is because, during these days, we count towards Matan Torah as like eagerly awaiting a beloved to arrive. Thus, most of this Masechta focuses on the passion for learning Torah and doing mitzvos. Moreover, this is the time of the harvest, which is a time when people follow their desires and pleasures. Therefore, to calm our passion and straighten our focus, we learn this Masechta of Avos filled with Mussar to evoke greater self-control and humility as we ask Hashem to protect us from straying after our physical passion during this time.