Are chemical bonds matter? So it recently blew my mind that chemical bonds have mass. And that a spring that's wound up similarly weights a little more. 
But there is a distinction between mass and matter. 
I believe that a chemical bond, even though it has mass, is not considered matter and is instead a form of energy. 
If I'm getting any of that wrong, I'd love to hear the rational. 
 A: Chemical bonds form when atomic orbitals of the nuclei to be bonded interact and form a molecular orbital.
The simplest case of bond formation is the formation of dihydrogen ($\mathrm{H_2}$) from 2 hydrogen atoms. The latter have (in the ground state) each one electron in a $1s$ atomic orbital and these orbitals then combine into a $\sigma$ molecular orbital, see schematic below:

What's most noteworthy is that the energy level of the $\sigma$ orbital is lower than that of the $1s$ orbital. Of course that means that energy is being released when hydrogen bonds (dihydrogen) forms.
To summarise, chemical bonds aren't really made of matter but small amounts of matter do convert to energy when they form.
And due to the mass-energy equivalence this also means that a very, very small amount of matter is converted to energy. The bond Enthalpy of $\mathrm{H_2}$ is about $-435\:\mathrm{kJ/mol}$ (a $\mathrm{mol}$ of hydrogen is about $2\:\mathrm{g}$), so you can work out just how little matter disappears though!
To summarise, chemical bonds aren't made of matter but small amounts of matter do convert to energy when chemical bonds form.
A: John Custer's comment is the best answer so far, so I'll wait for him to make it into an answer for now rather than copying it here.
Owing to the unified description of the "amount of content" of non quantum ground state systems through the system's total energy, the distinction between what a 1910s scientist would call matter and energy is no longer useful. Look up the article Matter on Wikipedia: the word is not used in science now, aside from in in niches applications and then it often clashes with everyday usage. For example General Relativity theorists call anything that contributes to the $0\,0$ stress-energy tensor "matter" but this includes all kinds of stuff that most lay people would not call "matter" in the everyday usage of the word. By this definition, the bond energy is most certainly "matter". But in general nowadays when we speak of anything aside from the quantum vacuum state (all "matter" and energy in, say, the 1910s usage) one needs to specify exactly what we have in a system: photons, electrons, muons, atoms arranged in chemical elements and so forth.
A: I have to say you have this backwards. Energy is released when atoms form bonds and therefore a decrease in mass takes place.
$E_{unbondedsystem}$ < $E_{bondedsystem}$ $therefore$ $M_{unbondedsystem}C^2$ < $M_{bondedsystem}C^2$
