Yes, "hard" in this context refers to worst-case complexity. In practice, lots of NP-hard problems can be solved, or approximately solved, in a reasonable amount of time. This is why traveling salesmen can exist, for instance.
Then it seems there should exist a more interesting category "really hard" which refers to average-case complexity. Yet I've never heard of such a complexity class?
Average-case complexity is studied quite a bit. There is some discussion on wikipedia [1]. Also see this paper, which discusses the average-case complexity of NP problems [2]
The reason why its not studied as much by theoretical computer science as is by algorithm-implementers, is that "average" is not uniquely defined. For instance, you can talk about the uniform distribution for a lot of problems, but not always. Suppose you want to investigate the complexity of solving certain types of equations, involving matrices as unknowns. How do you take the uniform distribution over matrices over the (infinite) real numbers? In principle, there are technical definitions you can make [3], and [2] has some results about this, but at the end of the day, there is no objectively superior way of doing it.
What this all leads to are two questions. (1) Would studying average-over-some-sampling-distribution-case complexity yield insights on what is computation? I don't know.
(2) Is it practically useful to classify problems in this way? Not really. Because in the real world, your distribution will rarely be uniform, so it is far better to optimize your algorithms for the distribution you do get.
Well, thanks for the Wikipedia link. It says average complexity is very important for sorting algorithms and cryptographic one-way functions. I found especially this part remarkable:
> Note that it is not desirable for the candidate function to be NP-complete since this would only guarantee that there is likely no efficient algorithm for solving the problem in the worst case; what we actually want is a guarantee that no efficient algorithm can solve the problem over random inputs (i.e. the average case). In fact, both the integer factorization and discrete log problems are in NP ∩ coNP, and are therefore not believed to be NP-complete.
Regarding this question:
> How do you take the uniform distribution over matrices over the (infinite) real numbers?
Well, naively, how about screw the real numbers and just take the uniform distribution over the, say, 64 bit floats, which are conveniently finite? At least if we use those in practice? But I guess this goes in the direction of "Because in the real world, your distribution will rarely be uniform", since the 64 bit floats are a subset of the real numbers, and any uniform distribution over the former is equivalent to a non-uniform distribution over the latter (probability 0 to any reals not in the 64 bit floats).
64-bit floats are a finite set. The solution of problems where the real numbers are replaced by the finite set is sometimes easier, but often much more difficult. The reason is that you can't do calculus over finite fields. For instance, taking the inverse of the natural log is efficient, but the inverse of the discrete log is a hard problem, and hence is used in cryptographic algorithms.
Beyond average case complexity, which is discussed below, there is also a rich research line on approximation algorithms. Interestingly, the effectiveness of approximation algorithms is not uniform across NP-Complete problems. There are problems where efficient approximation algorithms can be arbitrarily good and there are problems where if P!=NP then approximation algorithms have hard caps on their effectiveness. This is relevant for a lot of optimization tasks where you don't necessarily care about getting the best possible solution but would be happy getting pretty close.
Which would seem strange, since, intuitively, a hard worst-case means just that the problem is "possibly hard".