When finding correlation coefficient r by hand, as well as regression slope and intercept, you will see many calculations involving terminology you are unfamiliar with.

This is what is all means

sum(x) = all x's added together
sum(y) = all y's added together
sum(x^2) = square each x, then sum the total
sum(y^2) = square each y, then sum the total
sum(xy) = muitiply each xy value , then sum the total
(sumx)^2 = sum of the x's then square
(sumy)^2 = sum of the y's, then square
r^2 is just taking r and squaring it
y-bar is the average of the y's, which is sum(y)/n
x-bar is the average of the x's, which is sum(x)/n