What exactly is an integral? I understand it is the area under a curve but what does it pertain to?
The most common example in everyday life is perhaps the time integral of velocity is distance.
Every little bit of time dt, you go a distance dx = v dt.
If v happens to be constant the integral is easy
x = v t + Xi
where Xi is the x value at t = 0
That is just the area under a horizontal line.
BUT v(t) might be some more complicated function, for example
v(t) = Vi + a t
then
x = Xi + Vi t + (1/2) a t^2