The chemical elements most likely to form covalent bonds are those that share electrons, such as carbon, as opposed to those that take them from another element to form an ionic bond. In general, they are nonmetals with similar electronegativities. They are located toward the center of the periodic table, according to HowStuffWorks.
The purest form of a covalent bond exists in diatomic gases. Hydrogen, oxygen, nitrogen and the halogens all form these types of bonds. By sharing an electron they satisfy the octet rule for both atoms. Because the atoms have the same electronegativity, the shared electron has the same attraction to both.
The carbon-hydrogen bond organic materials require a covalent bond. These two elements form long chains that sometimes branch and have functional groups bonded to the chain. Breaking the covalent bond requires energy. In contrast, many ionic compounds readily dissociate when dissolved in water.
In polar covalent bonds, the two atoms continue to share the electron, but due to differences in electronegativity, one atom has more pull for the shared electron than the other. The draw is not strong enough to create an ionic situation. This is the type of bonding observed in water. As a result, water molecules have a positive end and a negative end, making it a polar solvent and giving it the ability to dissolve ionic compounds.