In the setting of Updike's story, the late 1950s, the A & P grocery stores were THE grocery store of small town USA. The name A & P came to be synonymous with "grocery store" in the way that Kleenex is synonymous with tissue. In small towns, no other store could compete with the A & P and it was where "everyone went." Thus, the A & P represented a part of Americana of the late 50s.
When the girls enter the A & P inappropriately dressed for the times, the manager is appalled and speaks to the girls. And, it is because of this store's iconclastic representation of the complacent American culture that the title has such significance. For, when Sammy quits the A & P, he breaks from the culture of his era. The manager, Lengel, whom Sammy describes as
thinking all these years the A & P was a great big dune and he was the head lifeguard,
concentrates "on giving the girls that sad Sunday-school-superintendent stare," and Sammy decides to defend the girls even when the punctilious manager says to Sammy,
Sammy, you don't want to do this to your Mom and Dad.
Like Holden Caulfield of "The Catcher in the Rye," Sammy refuses to be accepting of his social world and comprimise himself as the adults do:
But it seems to me that once you begin a gesture it's fatal not to go through with it.
Rather, he acts upon his impressions, even if it is against a cultural icon, the A & P grocery store:
When John Updike first published this story in 1961, A&P was the biggest grocery store chain in the United States. It was, at the time, selling over $5 billion worth of groceries every year, more than $1 billion more than its nearest competitor.
The chain had been around a long time -- its first grocery store opened in 1912. But before that, it had been a specialty shop for tea, coffee and spices. Beginning in 1869, the shop started using the name "Atlantic and Pacific Tea Company."
It is from that name -- "Atlantic and Pacific" that the initials came.