What does $O(\alpha(n))$ amortized time mean?
$\alpha$ indicates the Inverse Ackermann Function: $\alpha(n)$ is the number such that $A(\alpha(n), \alpha(n)) = n$.

In practice, $\alpha(n) \lt 5$ for any input less than about $^72$. So $O(\alpha(n))$ is basically $O(1)$. The difference only matters in theory, never really in practice. You'll find $O(\alpha(n))$ mostly when working with disjoint sets, which are pretty much the fastest data structures you'll ever use.

EDIT: $^nb$ indicates "tetration", that is, iterated exponents. It grows ridiculously quickly: $O(^n2)$ is worse than $O(n!)$.