In the fundamentalist, independent Baptist church, where I grew up, altar calls were at the end of every service. We had no altar. I suppose that means the term was figurative, but we were invited to come forward to lay it all on the altar without one.
The purpose was to announce having been saved in order to be baptized, to ask to move one’s membership from a church of like faith and practice, or to repent from certain sins. The altar call, often called the invitation in some similar churches, was right after the sermon and ran to the end of the service. In some cases, the people who went forward would be invited to take the microphone to speak to the congregation.
It began with a prayer, people being asked to have their heads bowed and their eyes shut, and then went into songs such as 'Just As I Am". If people went forward, they were met by the preacher who quietly talked to them. Once people stopped going forward, the preacher would ask people to look again and would announce, say, someone declaring he had been saved. In the context of that church, being saved meant that the person had an emotional moment wherein he decided to follow Christ and to submit to an immersion baptism. It was believed that a person who had a salvation experience could never lose salvation and had a guarantee of Heaven.
The more emotional moments were when people went forward to announce they were backsliders, a term for those who had not been strictly adhering to our faith’s requirements. That could be those who regularly skipped church, who smoked, who used alcoholic beverages, or who did other pursuits we considered too worldly. Those people would often speak to the congregation and would ask for prayer and assistance in “getting right with the Lord.”