Working in two and a half dimensions (2.5D) means you are cutting a part that has multiple flat features at varying depths. During a 2 ½ D cutting process, the Z axis positions itself to a depth where the X and Y axes interpolate to cut a feature. The Z axis then retracts so the X and Y axes can move to the start point of the next feature, which may be cut at a different Z depth than the last feature. Most simple CAM programs deal with 2.5D parts.
Working in three dimensions (3D) means that you have the ability to control at least three axes simultaneously. 3D contouring can then be accomplished by creating curves that use all three axes at once, like in a helical cut. You will most often need a full CAM program to create g-code files capable of performing 3D contouring.
Customers also have the option of using more than three axes in 3D part creation. 4th Axis milling usually describes situations where a rotary table is involved in the cutting process in addition to the X, Y and Z axes. The 4th axis can be used for full contouring with other axes, indexing, or flipping a part over. You will most often need a CAM program equipped with 4th axis capabilities to create a tool path for any part you plan on cutting using a rotary table. 5th axis adds one more dimension than the 4th axis. It is typically a rotary table on top of the 4th axis rotary table, otherwise known as a trunnion. It can also be a spindle that swivels. 5th axes are used for more intricate parts where undercuts are prevalent.