What Edwards is talking about when he uses these words in "Sinners in the Hands of an Angry God." (he actually says that he is speaking to people who “may keep up a form of religion in your families”) is people who have not undergone an actual conversion experience. These are people who have not been born again and who are, instead, just going through the rituals of religion without having the true feeling for it.
Edwards says that in order to have any hope of salvation you have to have embraced God on an emotional level. You have to have undergone a major change that has been brought about by God touching your heart. It is not enough to “keep up a form of religion.” In other words, it is not enough to go to church on Sunday and to say grace at meals and to otherwise engage in traditional displays of religious piety. Instead, you need to actually have changed your heart.
So, when Edwards says that some people “keep up a form of religion” he means that they just make an outward show of being religious without actually having experienced God touching them and changing their hearts.