The first step is to decide which format you need. Usually the first axis will be the X-axis. If you need simple X-Y motion then the next axis will be the Y-axis as in diagrams A and B. This would be X-Y format. You might instead need the second axis to be vertical; this is X-Z format.
If you need three axes of motion this could be X-Y-Z format where the vertical (Z) motion is the third axis as in diagrams C and D or X-Z-Y format where the vertical motion is the second axis as in diagrams E and F.
For the second and/or third axes you can choose whether the axis is fixed with a carriage that moves carrying the payload as in diagram A or carrying the next axis as in diagrams C and E, or you can have the carriage fixed so that the whole axis moves as in diagrams B, D and F. To clarify: in the case of a Y axis a moving carriage has the axis fixed and always occupying the workspace. The Y-axis is cantilevered and the carriage moves along it as in diagrams A and C. A moving axis has its carriage fixed to and carried by the X-axis and it moves in and out such that when fully retracted it extends backwards behind the robot as in diagrams B and D.
When a very long Y axis is required you need two X-axes, one at each end of the Y-axis to support it as in diagram G. The limit for a cantilevered axis is about 500mm depending on payload and expected acceleration.
The first axis, which is always called the X-axis (even though it may be mounted vertically) has to be fairly strong to support the weight of the other axes. Usual choice is the L-type which has dual crossed roller bearings. A K-type may optionally be used; it is fast and smooth but is harder to mount. The L-type is only ever used as the first axis.
Finally up to 3 rotary axes may fitted, pitch, yaw and/or roll.