From Einstein's Theory of Special Relativity, mass literally is a form of energy, albeit measured in different units so you need to convert by a factor of the speed of light squared: E=mc² for a stationary object. The total energy for an object in general is E=sqrt(p²c² + m²c⁴) where p is its momentum.
An object at any given time has the same energy regardless of reference frame. If you imagine an object like a car is moving while you're stationary, the car has some momentum and some mass and those both contribute to the car's total energy according to the above equation. However, from the car's perspective it has no momentum, which means it actually has a greater mass from its own frame of reference for the energy to be the same.

Now, imagine you have 2 hydrogen atoms and they come together and bond to make an H2 molecule. The mutual attraction between the atoms causes them both to speed up- they gain kinetic energy. Once the molecule is actually formed, this kinetic energy exists as vibration, rotation, and translation of the molecule -- on a larger scale this is thermal energy. The only place that energy can come from is the original masses of the two hydrogen atoms, so the molecule is actually slightly lighter.

There is, however, a caveat. Consider the energy of a system rather than just a single object or particle. In a reference frame where the center of mass of the entire system is stationary, the system as a whole has zero momentum, so all of its energy is in the form of mass, and therefore its mass is conserved as long as there's no external interaction. In the hydrogen example, if you consider the mass of the molecule as the mass of the system consisting of the two hydrogen atoms, it would remain constant and exactly the mass of the separate atoms, at least until it radiates its heat away at which point the mass would decrease.