Stronger bonds make more stable compounds both kinetically and thermodynamically. Kinetically because bonds must be broken for reactions to happen. Thermodynamically because bonds must be formed. A second factor is the reaction must be entropically favored. A third factor is the available energy thermal, photolytic or electrical must be sufficient to either not break bonds, stability, or to break bonds, reactivity.
Bonds are formed when electrons are shared between atoms, electrons are attracted to two or more nuclei. This can be a weak attraction such as intermolecular attractions; weak molecular bonds such as found in molecules such as $\ce{F2},$ $\ce{I2},$ $\ce{Li2};$ stronger bonds such as $\ce{C-H},$ $\ce{C-C},$ $\ce{C-O},$ $\ce{C-F},$ $\ce{C-Cl},$ $\ce{H-H},$ most ionic bonds and metallic bonds; and very strong bonds such as $\ce{C=O},$ $\ce{C=C},$ and the triple bonds in $\ce{N#N},$ $\ce{C#O},$ $\ce{HC#N},$ $\ce{HC#CH}.$ Double and triple bonds have unique properties because of their orbital structures.
All bonds are stable under low enough energy conditions or possibly a molecule in isolation. If energy is added: Heat, Photons, electricity, etc. Bonds will break. The energy to break sufficient bonds to cause a reaction is called the activation energy. The bonds will either reform or form new bonds. If the reaction is entropically favored by removal of energy stronger bonds will form and the reaction will proceed. The final products are determined by the strength of the bonds in the various compounds and the distribution of the energy involved.
Molecules with weak bonds are considered to be high energy; they tend to be reactive. Molecules with strong bonds are considered to be low energy; they are more stable. Reactions happen in the direction of forming stronger bonds and increased entropy. The latter is mostly manifested in the disposition of the energy evolved in the formation of the stronger bonds.