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An atom is neutral, which means it contains the same number of protons and electrons. The positive charge of the protons balances out the negative charge of the electrons and thus the overall net charge of the atom is zero.
Just to be confusing, this is not a rule that is set in stone. In some cases, atoms can acquire an electrical charge. An example we're all familiar with is good 'ole table salt, NaCl. The compound sodium chloride has a positively charged sodium atom and a negatively charged chloride atom. These atoms which are not neutral in charge are called ions.
Salt, NaCl, contains positively and negatively charged atoms called ions. (Image from here.)
It's a fairly simple concept for an atom to become an ion. They just have to gain or lose an electron. If an atom loses an electron it also loses a negative charge (and becomes positively charged). Positive ions are called cations. If an atom gains an electron, it also gains a negative charge (and becomes negatively charged) and is called an anion.
Let's consider NaCl in more detail. A sodium atom has 11 protons and 11 electrons. It has a neutral charge. The Na+ ion is formed from the loss of one electron. Because it has lost an electron, it has more protons than electrons; the overall charge of the sodium cation is positive. We've always found it useful to think of the "t" in cation as a + symbol. Similarly the Cl- ion is formed from the gain of one electron. Because it has gained an electron, it has more electrons than protons so the overall charge of the chloride anion is negative.
In case you're wondering, sodium chloride is called an ionic compound because it is formed from cations and anions.
Sodium Chloride, table salt, is an ionic compound. (Image from here.)
Truth is, an atom can actually lose or gain more than one electron. If an atom loses more than one electron, it becomes multiply positively or negatively charged. Some examples are Mg+2, Fe+3, and N-3.
Singly or multiply charged ions are called monoatomic ions. This means that ions like Mg+2 and Na+1 would both be included on the monatomic ion family holiday card.
There are ions in which two or more atoms combine to a net positively or net negatively charged species. These ions are called polyatomic ions and they include the hydroxide ion (OH-), the cyanide ion (CN-), and the ammonium ion (NH4+).
As added food for thought, it might be useful to know that metals tend to form cations, while nonmetals tend to form anions.
Chemistry cat, how we do love you.