What does excitation in an atom mean? This is not regarding the definition of excitation itself.
What I'm having trouble with is the difference between "electron excitation" and "atom excitation". When I electron inside the atom gets excited, it moves to a higher orbital. Now what about when an atom gets exited? "When an atom gets excited, ______" ?
You can't say the atom moves to a higher orbital, obviously.
What I'm thinking is something like: One of its electrons gets excited.
 A: In general, energy levels apply to the system1 (in your case the system of electron(s) and nucleus is the atom). So it is entirely appropriate to say that the atom is excited. 
It is only a few cases where it makes sense to factor the notion out and say that "this piece of the system" is excited. That works OK with hydrogen-like atoms because the nucleus is much more massive and can be thought of as approximately still if you are envisioning a classical orbit (not a good picture, really, but good enough for many purposes).
But once you have two or more electrons, the atom has more states available to it than are available to any single2 electron. So you can't use a enumeration of states for one electron to label the states of the atom.

1 In first year physics we often treat gravitational potential energy as belonging the the thing lifted in the classroom, but in fact it always belongs to the system of that object and the planet. The same rule applies here: states and their energies are associated with systems, not lone parts of the system.
2 Sloppy language there. The electrons are not distinguishable and so you can't say which electron was it that got excited. The question doesn't even make sense.
A: In nuclear physics, an exited atom is exited due to its nuclei spins being aligned in a energetically not minimized constellation. This can happen due to external energy intake or as a part of a radioactive decay where the mother nucleus' spin constellation is carried over but then nearly instantaneous changes in its daugther nucleus. The freed energy of the spin flipping is emitted as a photon at high energy (gamma radiation).
