bond energy
How tightly are two atoms held together? Bond energy answers that with a number: the energy you must supply to pull a bonded pair completely apart, from their resting distance out to infinity. It is the depth of the energy valley; a deep valley means a strong bond, a shallow valley a weak one. Equivalently, it is the energy released when the bond first forms.
Bond energies are usually quoted in electron-volts per bond or kilojoules per mole. Strong primary bonds run large: a carbon-carbon bond is about 3.6 eV (around 350 kJ/mol), a silicon-oxygen bond even more, which is why diamond and quartz are so hard and high-melting. Metallic bonds are moderate, and the secondary bonds are far weaker; van der Waals bonds are typically 0.01 to 0.1 eV, roughly a hundred times feebler, which is why waxes and molecular solids melt so easily. The bigger the bond energy, the more thermal jostling it takes to break the bonds apart into a liquid, so bond energy tracks the melting point closely.
Bond energy is the quantitative heart of how strong a material is at the atomic level. It sets not just melting and boiling points but also, through the depth-and-curvature of the same valley, stiffness and strength. It also explains chemistry's energy bookkeeping: a reaction releases energy when the bonds it forms are deeper (stronger) than the bonds it breaks. When you compare materials, bond energy is the first number to look at.
To melt and then boil a substance you must supply enough energy to overcome its bonds. Tungsten, with very strong metallic bonding, melts at 3422 degrees C; solid argon, held only by feeble van der Waals bonds (about 0.01 eV), melts at -189 degrees C. Same idea, hundred-fold difference in bond energy, thousands of degrees apart in melting point.
The depth of the energy valley sets melting point, stiffness, and strength together.
Do not confuse bond energy (depth of the valley, how strong) with bond length (position of the valley bottom, how far). A short bond is often but not always a strong one; the two are related through the curve but are distinct quantities.