An ion is something - an atom or particle of some sort - which has a charge. An electrical charge arises on a particle when the balance of its subatomic particles is lost; so instead of having the same number of protons and electrons, it has more of one than the other.
This is simply achieved by adding or taking away electrons from the outside of the particle, because the outside is the easiest bit to get to and less firmly stuck together than the nucleus or centre of the particle. If you lose one electron from the outside, then you'll have more protons than electrons, and end up with a positive charge of +1. If you gain an electron on the outside, you'll become negative with a charge of -1.
Most chemicals do not fly around in nature in the form of single atoms. They tend to form bonds. If both of the atoms that are forming a bond have an outer electron shell which is almost full, they will usually produced covalent bonds. However, if one is almost full and the other is almost empty, one will donate its electrons to the other to form an ionic bond.
This means that a lot of chemicals in nature are in the form of ions. This is particularly true of metals, e.g. sodium or potassium. In the body, these ions will commonly be referred to as electrolytes because, when they are dissolved in a solution, they help it to conduct electricity.
A cation is a positive ion. This means that it has lost at least one electron from the outside. Losing one electron means that the total number of protons in the nucleus will be bigger than the number of electrons around the outside. The overall (or 'net') charge is a positive one.
Cations do not need to have lost only one electron. The more electrons that are lost, the higher the number (e.g. +3). The number of electrons which can be lost usually depends on which atom you had to start with; for instance, a sodium cation is commonly a +1 ion. However, some atoms are able to vary the number of electrons they lose; for instance, iron is able to lose either 2 or 3 electrons, giving it a charge of +2 or +3.
Metals easily conduct electricity because they have less hold on their electrons. This means that a metal is more likely to lose an electron than to gain one. Therefore, when a metal atom becomes an ion, it is more likely to become a cation. The hydrogen ion is a unique example of a cation; it contains just the one proton, so it has more protons than electrons and carries a positive charge.
An anion is a negative ion. This means that is has gained at least one electron to its outer shell. Gaining one electron means the total number of electrons in the outer shell will be better than the number of protons in the nucleus. The net charge is a negative one.
Like with cations, anions can gain more than one electron, so the charge may be even less than -1. Oxygen, for example, tends to gain 2 electrons forming a charge of -2.
In nature, most acids are made up of an anion and a hydrogen ion; for instance, carbonic acid is made up of carbonate (an anion) plus a hydrogen ion. The acid is able to donate the hydrogen ion (which is what makes it an acid in the first place!), and is made up of the ionic bond between the negative anion and the positive hydrogen.