"Halogens" is the name for a family of chemical elements. They are found in group seven of the periodic table, the second column from the right. They are all electronically similar, having seven electrons in their outermost (valence) shell. Because this is only one electron away from having a full octet (eight) of electrons, which is a stable arrangement, the halogens have a high electronegativity. (Electronegativity is a comparative measure of how readily an atom will accept another electron.) Fluorine, the lightest halogen, has the highest electronegativity of any element. The electronegativity values decrease as you follow down the column, but still remain substantial compared to other elements.
In the elemental form, the halogens exist as diatomic molecules. "Diatomic" simply means two atoms bound together. Because they each require only one more electron to complete their octet, a single covalent bond joins the two atoms together. (A covalent bond is a sharing of one electron from each atom, so each atom effectively gains the one electron it lacked.) The lightest two halogens, fluorine and chlorine, exist as gasses at room temperature. Bromine exists as a liquid (mercury is the only other naturally occurring liquid element at room temperature), and iodine and astatine (which is radioactive) are solids.
Halogens readily form monatomic (one atom) ions by "stealing" an electron away from metals. This process is so energetically favorable that explosions can result if the reaction between halogen and metal is not controlled. When the halogen atom gains its own electron, it does not need to share with another atom any longer. Instead, it exists on its own as a charged particle (an ion). Halogens form monatomic ions with a charge of -1, because they only have one extra electron. Like all negative ions, halogen ions are attracted by positively charged ions and can form crystal lattices - solid structures made by regularly spaced positive and negative ions (often called cations and anions, respectively). The positive ions are generally formed from metals (which have lost electrons, often to halogens). The most familiar crystal structure made from a halogen is table salt - sodium chloride.
Halogens also make binary acids when paired with hydrogen. (Binary means two - referring to the fact that there are just two atoms paired together.) In water, these compounds dissociate - break up into their ions. All are strong acids, meaning that they break up nearly completely, save for fluorine's pairing, which only partially dissociates, marking it as a weak acid. (The release of the hydrogen ion is what makes a solution acidic.) The acids are named according to which halogen they are made from, with the prefix "hydro-" to indicate they are paired with hydrogen. Thus you see hydrochloric acid, HCl, which is found in your stomach, or hydrofluoric acid, HF, which is used to etch glass.
Halogens are commonly found in other compounds and ions too. Because they have that high electronegativity, they will react with just about anything, and so they play a part throughout chemistry. Many polyatomic ions use halogens as a central atom. Halogens are easily added into organic compounds, as they will grab the extra electrons found in double bonds between carbons. Many poisons (like DDT) are organic compounds which contain halogens. The body's normal chemistry does not include halogenated compounds. Science has exploited this fact to produce molecules that are useful as well. A modern example is the sweetener "Splenda/Sucralose" which is made by adding chlorine to sugar molecules, rendering them indigestible by the body.
If we were to anthropomorphize halogens, we could say that their chemistry is overwhelmingly a result of their strong desire for just one more electron, and that they are willing to react in pretty much any manner to get it. Remembering that will go a long way towards explaining everything that they do.