Metal and Vulkan are low-level 3D graphics APIs, in much the same way that OpenGL and Direct3D are 3D graphics APIs. Metal is Apple's API, usable on iOS and MacOS. Vulkan was developed by the Khronos Group, who also oversees the various forms of OpenGL. Vulkan is directly usable on all non-Apple platforms, and there is the MoltenVK project which provides a translation layer from Vulkan to Metal (since the APIs are not that different), thus allowing Vulkan applications to run even on Apple platforms.
Metal, Vulkan, and Direct3D v12 represent a dramatic departure API-wise from prior low-level graphics APIs. They use a command-buffer-based interface, where you record rendering commands into a command buffer object, then execute those commands at some later period. The more immediate style of APIs like OpenGL and D3D-pre-12 have no intermediary command buffer. When you call a rendering function, it is given to the GPU "right then" and that's the end of it.
These styles of API are important because they allow threaded creation of rendering commands. In the immediate APIs, you can only effectively have one thread issuing rendering commands. In command-buffer APIs, multiple threads can build multiple CBs simultaneously, and then one thread submits them all as a single bunch of work. This allows better utilization of CPU resources for CPUs with lots of cores.