The private pilot license allows a pilot to operate a SEL aircraft under normal VFR conditions. It is the certification held by the majority of pilots and the first step to becoming a commercial pilot.
The Multi-Engine license is an "add-on" to the single-engine license and allows a pilot to fly an aircraft with more than one engine. This license broadens a pilot's horizon in regards to speed and power.
The CFII is the next step after completing the CFI training. The CFII allows a flight instructor to instruct instrument students and an MEI allows an instructor to give his students multi-engine instruction.