From what I have learned asymptotically tight bound means that it is bound from above and below as in theta notation.But what does asymptotically tight upper bound mean for Big-O notation?
What is an asymptotically tight upper bound?
asymptotics
Saying that a big-O bound is asymptotically tight basically means that the author should have written $\Theta(-)$.  For example, $O(x^2)$ means that it's no more than some constant times $x^2$ for all large enough $x$; asymptotically tight means it really is some constant times $x^2$ for large enough $x$ and not, say, some constant times $x^{1.999}$.