This is a Heyawake ("divided rooms") puzzle.
Rules of Heyawake:
Shade some cells of the grid.
Shaded cells cannot be orthogonally adjacent; unshaded cells must be orthogonally connected.
There cannot be a horizontal or vertical line of unshaded cells that passes through two borders.
If a number is in a room, there must be exactly that many shaded cells in that room.