There are three main types of chemical bonds: ionic, covalent, and metallic. Each are between different kinds of atoms, and have general properties. Components, properties, and discussions of degree are all below.
Ionic bonds are formed between a nonmetal and a metal. A bond between two atoms is an ionic bond if the difference of their electronegativities is between 1.7 and 4. That is to say, if one atom has an electronegativity that is more than 1.7 greater than the atom its bonded to, the bond is ionic. Because the electronegativity is so great, electrons are completely gained or lost.
Ionic bonds bonds, generally, are very hard, and have a high melting point. However, most ionic compounds are soluble. In a solid form they don't conduct electricity, but when they are dissolved or liquefied they generally do.
Example: NaCl: salt
Covalent bonds are formed between two nonmetals. A bond is a covalent bond if the difference between the electronegativities of the two atoms is between 0 and 1.7. There are two types of covalent bonds: polar and nonpolar. If the difference in electronegativity is between 0 and 0.3, it is nonpolar, if it is between 0.3 and 1.7, then it is polar. Polar simply means that electrons are unequally shared, whereas in nonpolar bonds electrons are shared equally.
Covalent bonds can be either soluble or not soluble, and generally have low melting points. They are, in general, never conductive.
Example: O2: oxygen
Metallic bonds occur between two metals. These compounds are shiny, not soluble, and have high melting points. They are always conductive.
Example: Cu: copper