“Lattice confinement” refers to the lattice structure formed by the atoms making up a piece of solid metal. The NASA group used samples of erbium and titanium for their experiments. Under high pressure, a sample was “loaded” with deuterium gas, an isotope of hydrogen with one proton and one neutron. The metal confines the deuterium nuclei, called deuterons, until it’s time for fusion. post 
Key to this process is an effect called electron screening, or the shielding effect. Even with very energetic deuterons hurtling around, the Coulomb barrier can still be enough to prevent fusion. But the lattice helps again. “The electrons in the metal lattice form a screen around the stationary deuteron,” says Benyo. The electrons’ negative charge shields the energetic deuteron from the repulsive effects of the target deuteron’s positive charge until the nuclei are very close, maximizing the amount of energy that can be used to fuse.
Nuclear fusion reactions in deuterated metals: We show that neutrons are far more efficient than
energetic charged particles, such as light particles (e–, e+) or heavy particles (p, d, α) in transferring kinetic energy to fuel nuclei (D) to initiate fusion processes. pdf
Novel nuclear reactions observed in bremsstrahlung-irradiated deuterated metals. abstract