A delay differential equation is a bit like a regular equation, but it's a little more complicated. In a regular equation, you have an unknown number (or numbers) that you are trying to figure out. In a delay differential equation, there's still an unknown number, but it depends on how things were in the past. For example, if you want to figure out how hot it's going to be tomorrow, you need to take into account the temperature from today, the temperature from yesterday, and maybe the temperature from the day before. A delay differential equation can help you figure out these past temperatures and the temperature for tomorrow.