What does it mean when someone says a numerical scheme or a time integration algorithm is "energy conserving". How can a numerical scheme "gain" or "lose" or "conserve" energy apart from the numerical diffusion that is inherent.
"They" are probably talking about symplectic integrators.
Most numerical integrators for (partial) differential equations do not specifically consider the energy of the system; they are generic integrators capable of solving any set of DEs, and not all DE's have a concept like "energy".
When these are applied to a classical dynamics problem concerning some conservative system, one of their error modes tends to be that that system's energy is not conserved.
Symplectic integrators are specific to classical dynamics and are designed with conservative systems in mind. They eliminate this particular error mode, and guarantee that the system's energy will be conserved.
No numerical scheme is absolutely perfect; they of course have other error modes, such as round-off error (the one you mention).