Why do atoms have no electric charge even though most of their particles have charges?

Explanation:
An atom is made up of a nucleus surrounded by electrons, in very basic terms. The nucleus is made up of protons and neutrons, and so the three basic particles in an atom are protons, neutrons and electrons. An electron has a negative charge $(-1)$, a proton has a positive charge $(+1)$, and a neutron has no charge (0).
Every atom has a specific number of particles within it. The number of protons in the nucleus of an atom defines what element it will be, the number of neutrons decides what isotope it is, and the number of electrons in an atom, is always equal to the number of protons (an ion is when an atom loses or gains electrons, but that's irrelevant).
Take the example of a carbon atom. It has six protons, six electrons, and six neutrons. The number of protons it has is equal to its atomic number (or proton number), which defines its place on the periodic table. If you add together the charges of each particle in the atom you get:
$6(+1)+6(-1)+6(0)=6-6=0$
And thus the atom has no overall charge, despite the individual particles having charge. Another example, chlorine (17 protons, 17 electrons, 18 neutrons):
$17(+1)+17(-1)+18(0)=17-17=0$