Warning: Episode 5 spoilers ahead! On The Handmaid's Tale, Offred's shopping companion Ofglen (who is renamed Ofsteven in "Faithful," but is really named Emily) always seemed to know more than a handmaid should. She had information about what was going on in other cities and a network of people whom she could ask questions. It was clear that she was working with some kind of secret organization, but it wasn't until Episode 5 that that group received a name: Mayday. But what is Mayday on The Handmaid's Tale?
Mayday is the name of the secret resistance in Gilead. Not much is known about them aside from the fact that they're "fighting back," in Emily's words. They have access to information and have placed spies throughout Gilead; Emily certainly couldn't have been the only member of the resistance hiding in plain sight. But because Gilead is a nation under such heavy surveillance, where everyone is constantly being watched and encouraged to inform on those who misbehave, their work must be conducted very carefully. After Emily was put on trial and genitally mutilated as punishment for her relationship with a Martha, she was too dangerous to continue her Mayday work. People in the resistance have to be incredibly careful to avoid capture.
The term "mayday" gained popularity in the 1920s as an alternative to S.O.S., because, according to the Oxford English Dictionary, there was some trouble "distinguishing the letter 'S' by telephone." Both terms were used as radio distress signals for ships and planes. As mentioned by Offred in "Faithful," mayday came from an anglicized version of the French word m'aidez, meaning "help me."
Emily encouraged Offred to find Mayday and join forces with them, but was not able to relay how Offred could find them. In the Margaret Atwood novel the show is based on, Offred does not become personally involved in the resistance, though they are a recurring thread throughout the book. Ofglen (whose real name is unknown in the book) is a member of Mayday who takes her own life when the connection is discovered. And there is an indication at the very end of the novel that Nick is a member of Mayday, too. He appears to help Offred escape from Gilead, though the novel doesn't technically confirm this.
Mayday isn't just the name of their organization, it's a code word that allows them to know when they're speaking with a kindred spirit. The knowledge that people are fighting back, even if they have to do so slowly and carefully and quietly, provides hope that not all is lost in Gilead.