The word "excitation" embraces the concept of quantization of a field.
The first excitation of a field is one quantum (unit) of the field.
The second excitation of a field are two quanta (units) of the field.
It is valid for electromagnetic, weak, strong-interacting and possibly also the gravitational field (a consistent theory of the latter is still lacking). It is the concept to make compatible an a-priori continuous field with its actually discontinuous quanta.
The first excitation of a field is the smallest non-zero field (but there are exceptions, I come to that below) that can exist according to quantum field theory. A strong field, however, consists of a myriad of its quanta.
It is this weird behaviour of nature that a field, for instance an electromagnetic field, when it is scattered for instance at a slit or even a double-slit behaves like a wave and when it impinges on the detector behind the slit it is registered as a series of quanta.
The photon is the smallest unit of an electromagnetic field. It cannot be split in two parts to create an even smaller unit of the electromagnetic field. From the point of view of a classical continous field this cannot be understood, however, this is how nature really behaves, it has quantum properties.
However, the quantum nature is not always noticable.
Because one quantum, although being only one, can have different energies. So a slowly changing field will consist of low-energy quanta, whereas a fast changing field of high-energy quanta.
The dectection, however, of single low-energy quanta is very difficult, so the overall behaviour of such a system will be observed as a continuous field. However, high energy quanta can be detected as a series of single quanta, because the energy change in the detector is large enough to be detectable.
Now let's come to the famous Higgs-boson. The concept is applied exactly in the same way, one Higgs-boson is first excitation of the Higgs field, two Higgs-bosons are the second excitation of the Higgs field and so on. A "strong" dynamical (changing in time) Higgs field consists of myriad of Higgs-bosons.
It has, however, a property that distinguishes it from other quantum fields.
Whereas in the majority (for instance the electromagnetic/photon field) of fields if there are no quanta, the field value is zero. For instance if there are no photons, there is no electromagnetic field. Its field value is zero, i. e. electric and magnetic field strength are zero. This is not true for the Higgs-field. In space there might be no Higgs-boson at all, but the Higgs-field is non-zero. This non-zero field is not quantized. It cannot be considered as myriad of Higgs-bosons. This is what is meant with the vacuum expectation value of the Higgs field is non-zero. It cannot be "damped". It is always there. Higgs-quanta decay, they disappear, but the Higgs-field without any Higgs quanta persists.
Part of the question is "How do we get an excitation?". Everywhere where is an interaction. One of the simplest case is the light bulb. High temperature make the electrons in the wire get to higher energy levels from which they return to their original energy levels by the emission of light, or differently said, by the emission of light quanta, photons. It is an interaction of the electrons with the electromagnetic field. Photons have no mass, so there is no energy threshold necessary to create them, the interaction can be very weak.
For Higgs-quanta it needs at least 125GeV to produce them as they have a large mass. So particles with sufficient energies have to be smashed to create them.
