What is conjugate gradient descent?
What does this sentence mean? 


It means that the next vector should be perpendicular to all the previous ones with respect to a matrix. It's like how the natural basis vectors are perpendicular to each other, with the added twist of a matrix:

$\mathrm {x^T A y} = 0$ instead of $\mathrm{x^T y} = 0$


  And what is line search mentioned in the webpage?


Line search is an optimization method that involves guessing how far along a given direction (i.e., along a line) one should move to best reach the local minimum.