The other day, I set my thermostat to 70 degrees. It was 65 degrees in the house. After a short time I noticed that the temperature in the house wasn't changing and no air was coming out of the vent, so I checked the burner. The pilot was still going and the burner was set to on, not pilot. The burner should have ignited but didn't. Then I noticed a black wire nor connected at one end. How can I figure out how it connects properly so I can test the burner and see if it ignites?

You need a schematic for your particular furnace, with model and make serial numbers. Or a trusted reasonable technician.

Any further guidance require full ID of the furnace.