Ionic bonding: a type of chemical bonding between the atoms of two different elements. The two elements are introduced to each other electrostatically, exchange electrons and become attracted, thus bonding to each other. An example of this is sodium fluoride. Sodium has one more electron in its outer shell than fluorine does. Electrolysis causes an electron to leave the outer shell of the sodium ion and enter the outer shell of the fluorine ion. This exchange of electrons leave the sodium ion positively charged and the fluorine ion negatively charged. Opposites attract, so they join together and form a new compound.
Covalent bonding: a type of chemical bonding between two atoms of similar negativity, thus providing a stable bond in which they share a pair of electrons. Methane is an example of this type of bonding. Both hydrogen and carbon atoms have 1 electron in their outer shell. When electrostatically introduced, they crowd together, overlap, and share a pair of electrons between them. It's sort of like three people sharing two chairs or a Venn diagram with overlapping areas.
Hydrogen bonding: a type of bonding between a hydrogen atom and another electronegatively charged atom such as fluorine, oxygen, or nitrogen. Water (H2O) is a classic example of this type of bond. Two hydrogen atoms basically sandwich an oxygen atom between them. The oxygen has a negative charge and the hydrogen a positive charge. Therefore, every oxygen atom is attracted to a hydrogen atom, and the molecules clump together in a cohesive liquid form. This bond is so strong that it takes a relatively high heat to break the molecules apart, thus explaining why water has such a high boiling point in comparison to other compounds.