The camera coordinates are the same as image coordinates. So You have x axe pointing in the right side from the camera, y axe pointing down, and z pointing in the direction camera is faced. This is a clockwise axe system, and the same would apply to the chessboard, so if You specified the origin in, lets say, upper right corner of the chessboard, x axe goes along the longer side to the right and y along shorter side of the chessboard, z axe would be pointing downward, to the ground.
Solve PnP outputs the translation in the same units as the units in which You specified the length of chessboard fields, but it might also use units specified in camera calibration, as it uses the camera matrix.
Tvec points to the origin of the world coordinates in which You placed the calibration object. So if You placed the first object point in (0,0), thats where tvec will point to.