The area law says that the entanglement of any part of a system with the rest of of the system scales like the boundary (the "surface area") of the region. E.g., in a one-dimensional chain, the entanglement of a contiguous block with the rest should be bounded by a constant, and in 2D, the entanglement of e.g. a square region with the rest should scale like the linear size of this square.
The area law is a property which is proven to be satisfied by ground states of local gapped Hamiltonians in one dimension (see arXiv:0705.2024 and arXiv:1301.1162), and the corresponding statement is believed to be true in two dimensions. However, even for systems without a gap the area law is only mildly violated (in that the entanglement does not grow like the volume).
Local Hamiltonian refers to the fact that the Hamiltonian is a sum of terms each of which only acts on a small number of closeby spins, e.g. nearest neighbors on a lattice.