One of the lights on your Christmas tree burns out and this causes the rest of the lights connected to that strand to go dark. How are the lights in that strand connected?

Lights connected in series form a chain with each light connecting to adjacent lights via one wire. Therefore, if one of the lights burns out and breaks the series, none of the other lights will receive power and they'll go dark. Lights connected in parallel each connect to the positive and negative nodes of the power source and would not go dark if one of their neighbors burnt out.