The law of conservation of atomic mass says that in a chemical reaction the sum of atomic masses that enters into a reaction equals the sum of atomic masses that is produced in the reaction. In other words, the left member of a chemical equation equals the right member of the equation. This law has been proved by Lavoisier, a French chemist considered to be the founder of modern chemistry. Unfortunately, he was executed during French revolution.
In a nuclear reaction the above law is not valid any more because the sum of particles' masses that enters into a nuclear reaction is far higher than the sum of particles' masses that are produced by the nuclear reaction (nuclear fission not nuclear fusion). This phenomenon is called mass defect or also known as mass deficiency.
Mass is a measure of inertia, it is not a measure of substance, as many believe. That mass can not disappear just like that, it is carried by some particles with very high speed. These particles are called neutrinos. The mass defect is not a confutation of Lavoisier law because the law refers strictly to chemical reaction and atoms while the nuclear reactions works with subatomic particles.
Mass is relative and depends on the speed of the object. When the speed is higher, the mass is higher. Mass is not weight. Weight is also relative and it depends on the intensity of the gravitational field where the object exists. Just the quantity of substance is absolute.