This all breaks down to movement of the machining center. Think about cartesian coordinates. The x-axis is one axis, y-axis another, and z-axis another. That makes three axes. This is the most common situation. A machine that moves along the X, Y, and Z axes is generally a three axis machine.
Now lets talk about exceptions. A machine that can move in three axes but only two simultaneously, lets say x and y in this case, is considered a 2.5 axes machine. Again, technically three axes, but only two can be utilized at one time.
There are also four and five axis machines.
A four axis machine generally has X, Y and Z axes and a rotary axis usually called the 'C-axis'.
Here again, some machines can move in all four axes simultaneously and are true 4-axis machines.
Three axis machines are sometimes retrofitted with a rotary c-axis and are usually considered 3.5 axis machines because the 4th axis cannot move simultaneously with the other three axes.
This same situation exists for five axis machines. Some are true 5 axis machines. These are often used for very complex work such as impeller manufacturing and other areas of pump design. There are also 4.5 axis machines that have a axis added that are not part of the original design intent of the machine.
One more thing. Lathes are generally 2-axis machines, X and Z. Z-axis is parallel to the axis of rotation and X-axis is perpendicular.
I must take exception to some statements in the above.
The Z axis is always parallel to the spindle axis. However, the "A" "B" and "C" denote axes having rotation about the X Y and Z axes respectively.
So it would be relatively unusual to have a machine which had only X Y Z and C axes - in fact it would be almost pointless for a machining center, although you do find lathes which have an additional precision positioning capability for the spindle, which is therefore denoted the C axis. The most common machining center configuration is X Y Z and B (the B denoting the axis of a rotary table with its axis parallel to the Y axis - which is more often than not the vertical axis). A lot of people are unaware of this convention for axis lettering - there is an ISO standard which covers it. In this standard, you will find further conventions relating to additional axes, multiple parallel axes, and which letters cannot be used etc.
