If your part is any sort of shape other than cylindrical, or lacks rotational symmetry about some imaginary axis through the part body, then a CNC mill is likely the machine for you. With milling, the stock that the part will be cut from is usually a six-sided bar or sheet of material (though it can also be a cylindrical rod). This stock is fixtured to the bed of the machine and the features of the part are cut using a rotating tool such as a drill or endmill.
With a 3-axis machine, the rotating tool mentioned above can translate relative to the part in the X, Y, and Z axes (thus the name), but is fixed in a vertical (or horizontal, depending on the machine) orientation. With modern CNC mills, the tool can also move in any combination of the 3 axes simultaneously.
A 5-axis machine is capable of the same movements as a 3-axis machine, but with added rotation about the X and Y axes. It is usually the bed that rotates rather than the tool-head, though both types of machines do exist. This added rotation allows for much more freedom when machining each setup.