What is the difference between a tight Big $O$ bound, a tight Big $\Omega$ bound and a Big $\Theta$ bound?
$O$ and $\Theta$ refer to different properties of some function. In a sense, we know more of a function when we can describe it with $\Theta$ than with $O$. 

However, usage of $O$ or $\Theta$ doesn't imply tightness of a bound. You have some function $f(n)$ that is a bound which you could either describe with $f(n)\in O(g(n))$ or $f(n)\in \Theta(g(n))$, but whether the bound is tight depends on the actual bound $f$.

So, as a tight bound means simply that there is no 'better' bound for all cases, a tight asymptotic bound (either in $O$ or $\Theta$, or even $\Omega$), simply means that there is no better asymptotic description of a bound.

An example of a tight bound with only $O$ is the time complexity of finding an element in an unordered list of size $n$. This complexity is bounded by $O(n)$ and this is tight. The complexity is not $\Theta(n)$, as there are cases when we find the element without scanning the entire list. (The asymptotic complexity of the worst case is $\Theta(n)$, though)

For more reading on bounds with Landau ('big $O$') notation, see the reference question How does one know which notation of time complexity analysis to use? and its answers.