A Chemical Bond is the force that attracts atoms together, allowing the formation of chemical compounds made of two or more atomic elements. Ionic and hydrogen bonds are both types of chemical bonds.
Ionic Bonds are formed by two oppositely charged ions attracting together between metal and non-metal ions. All ionic bonds have one or more covalent bonds, or sharing of electrons, which aids in the attraction by allowing the non-metal ion to gain a higher negative charge. Ionic bonding is relatively stable but is usually easily dissolved by other compounds, such as salt in water; the strength of the bond is in its stability.
Hydrogen Bonds are the specific bonding of hydrogen with other atoms or molecules. When hydrogen bonds with another atom, such as oxygen, it then can link up with other molecules without losing the molecular coherence; thus, water molecules can form a liquid, all molecules attached together, without losing the molecular structure of H2O.
Because they share electrons, the attractive force of an ironic bond is more powerful than the general attraction of a hydrogen bond. Hydrogen bonds need less energy to break, while ionic bonds usually need a catalyst with more energy to break the chemical bonding.