Nonvolatile charge-trap memory plays an important role in many modern electronics technologies, from portable electronic systems to large-scale data centers. Conventional charge-trap memory devices typically work with fixed channel carrier polarity and device characteristics. However, many emerging applications in reconfigurable electronics and neuromorphic computing require dynamically tunable properties in their electronic device components that can lead to enhanced circuit versatility and system functionalities. Here, we demonstrate an ambipolar black phosphorus (BP) charge-trap memory device with dynamically reconfigurable and polarity-reversible memory behavior. This BP memory device shows versatile memory properties subject to electrostatic bias. Not only the programmed/erased state current ratio can be continuously tuned by the back-gate bias, but also the polarity of the carriers in the BP channel can be reversibly switched between electron- and hole-dominated conductions, resulting in the erased and programmed states exhibiting interchangeable high and low current levels. The BP memory also shows four different memory states and, hence, 2-bit per cell data storage for both n-type and p-type channel conductions, demonstrating the multilevel cell storage capability in a layered material based memory device. The BP memory device with a high mobility and tunable programmed/erased state current ratio and highly reconfigurable device characteristics can offer adaptable memory device properties for many emerging applications in electronics technology, such as neuromorphic computing, data-adaptive energy efficient memory, and dynamically reconfigurable digital circuits.