Matrix product state (MPS) is a "method" to write down an area-law-entangled many-body quantum state in 1D. Depending on the dimensionality and size of the system an MPS can more or less efficiently approximate the full state.
It's important to understand that while it may be possible to describe the full many-body state accurately using much less information than the full state has ("much" means exponentially less), it is not obvious that it is possible to do it within an exponentially more efficient algorithm than exact diagonalization.
DMRG is an efficient method to perform that optimization. For example, it can target ground states of gapped 1D systems accurately. However, that only scratches the surface of its applicability. It's important to state here that DMRG can target general states that can be well approximated by MPS and not just ground states, however, to do that you need to do Shift-Inversion with some iterative $Ax=b$ solver. It is also important to state that DMRG may not be the most efficient algorithm, but it is the most common algorithm to optimize MPS.