Question

What is the linear regression line?

Answer 1

It's the line that gives a closest fit between to variables if there is supposed to be a linear correlation.

Example:
In my job as a teacher I had the feeling that students that scored good in Maths also scored good in Physics and vice versa.
So I made a scatterplot on a chart in Excel, where x=Maths and y=Physics, where each student was represent by a dot.
I noticed that the collection of points looked like a sigar-shape in stead of being all over the place (the latter would mean No correlation at all).

And then I did two things:
(1) I had the correlation coefficient calculated (which was high)
(2) I had the "line of best fit" drawn
The latter one is the regression line, and you can even have an equation attached to it.
From this you may make a more or less reasonable prediction of one score from the other, depending on how good the correlation is (correlation is another subject).

Remark:
There are a lot of 'buts' and 'ifs'. For one thing you have to be reasonably sure the correlation is linear.