This phrase can be interpreted on a few different “levels.”
In the context of chemistry, it refers to the fact that chemical reactions only rearrange atoms. It means that the number and types of atoms is equal on both sides of a reaction. An atom can never be created, nor destroyed, in a chemical reaction. The atoms are merely rearranged.
Let’s say that you have some methane (CH4). Then, you set the methane on fire. The reaction is as follows:
CH4 + 2O2 = CO2 + 2H2O
(methane + 2 oxygen gas = carbon dioxide + 2 water).
If you add of the mass of the methane/oxygen molecules, and compare it to the combined mass of the carbon dioxide/water molecules, you will find that the masses on both sides are equal.
The next level would be in the case of nuclear physics. The equation E=mc^2 comes into play. As nuclear bombs have demonstrated, it is indeed possible to “destroy” a small percentage of matter during a nuclear reaction. When matter is destroyed, it releases energy.
The reason that nuclear weapons release energy is because of something called “mass defect.” Basically, protons and neutrons can be arranged in different ways, giving way to the different elements with different potential energies. Potential energy manifests itself as mass, so an atom will weigh ever-so-slightly less than the sum of its parts.
A common nuclear reaction is:
U235 + n = Kr92 + 141Ba + 3n
(the n’s are neutrons)
In this example, if you add the weight of the left hand side, you will find that it will weight more than the right hand side. Where does that mass go? If you count the particles (protons/neutrons) on both sides, you will find that the sum has not changed.
When the uranium splits, it lowers its potential energy, reducing the mass of the result. This lost mass is converted into energy. Since it is multiplied by a factor of c^2 (the speed of light squared), a small amount of mass creates a lot of energy.
On an even deeper level, the relationship between mass and energy can become very blurry. In subatomic physics, mass is easily turned into energy, and vice-versa. In fact, when it comes to subatomic particles, mass and energy are the same thing.