(difference in an ion's electron count versus proton count)
An ion is an atom or molecule that has more or less electrons than
protons, as opposed to a neutral atom in which the numbers match.
The term state of ionization (or, more briefly, just ionization)
is used to characterize this surplus or deficit, such
as +2 for "missing two electrons".
In such a case, fully ionized means missing all the electrons,
and neutral means missing none.
In astrophysics, the following terminology is often used:
the chemical symbol followed by I means neutral,
II means singly ionized, III for doubly ionized, etc.,
e.g., CI, CII, CIII for carbon
at these three states of ionization.
The more common symbol in other sciences is (e.g., for hydrogen),
H+ for singly ionized, H++
for doubly ionized, etc.
Atoms or molecules with extra electrons (i.e., more than their
protons) are also termed ions: I presume this is less common,
and generally, such extra electrons are very readily freed.
In context, state of ionization (and the term fully ionized)
may be used to mean the degree of ionization,
referring to a number of atoms, characterizing how many
are ions and how many are at each state.
For a collection of atoms/molecules, e.g., a gas, the percentages
of the gas at each state of ionization is of interest,
because of its relation to temperature and to the production
of the spectral lines, allowing the study of spectral lines
to be used to determine temperature, through the Saha equation.