How many electrons, and how many protons, does an atom have?

If an atom has a charge (i.e. # X^"+1" or X^-1#) then it is considered to be an ION. The charge of an ion tells you how many more or less electrons it has compared to it's original atomic number.

An electron has a negative (-1) charge and so an ion with a -1 charge will have gained an electron and so it's total number of electrons will now be it's atomic number plus one.

If an ion has a positive charge it has lost electrons. So a +1 ion charge will mean the ions total number of electrons will be it's original atomic number minus one.

...specifically, what is #Z#, #"the atomic number"#, that defines the number of positively charged, massive particles, i.e. protons, in that element's nucleus?

Let us take for example...#Z=8#, a quick captain at the Periodic Table tells us that if #Z=8#, then we got the element oxygen...(oxygen like most elemental gases, certainly the gases with any chemistry, is binuclear, i.e. #O_2#..) but for the moment let us consider an isolated oxygen ATOM.

If we gots an electrically neutral atom, then if there are the 8 positively charged, massive particles, i.e. protons, and there are by SPECIFICATION, THERE MUST be 8 electrons in the NEUTRAL ATOM....and these electrons are conceived to whizz about the nuclear core in all sorts of fancy orbits, with fancy energies. And because the element lies towards the right of the Periodic Table as we face it#"....."#its high nuclear charge tends to make oxygen oxidizing...i.e. it tends to pick up electrons from OTHER elements, and itself becoming reduced....the which process we could represent symbolically...

#O + 2e^(-) rarr O^(2-)#

Its propensity to pick up electrons, to become reduced, may also be rationalized on the basis that #O^(2-)# is ISOELECTRONIC with the proximate Noble Gas, neon, which has a stable electronic arrangement.

Confused yet? What I am saying is that you should refer to the Periodic Table in all your consideration of chemistry. And the Periodic Table is always an allowed examination material. You still have to learn how to use it effectively. Given the name of the atom, the Table gives you #Z#, and #Z# also gives you the number of electrons in the NEUTRAL species.