Some variant definitions define it as the number of time steps taken by the machine instead. Wikipedia talks about their relationship, but no patience right now.
This problem is theoretically interesting because many important mathematical conjectures have been proved to be equivalent to whether a given Turing machine halts or not, this is notably the case for:
Therefore, if we were able to compute , we would be able to prove those conjectures automatically, by letting the machine run up to , and if it hadn't halted by then, we would know that it would never halt.
Of course, in practice, is generally uncomputable, so we will never know it. And furthermore, even if it were, it would take a lot longer than the age of the universe to compute any of it, so it would be useless.
However, philosophically speaking at least, the number of states of the equivalent Turing machine gives us a philosophical idea of the complexity of the problem.