What is the difference between SMOH and SFOH?
The life of an aircraft engine is measured in hours of operation since some major service was done to it. When looking at ads for planes there are three regulatory terms that you'll see in engine descriptions:


New (TSN)
This one is pretty self explanatory - A brand new engine, produced with all brand new parts that comes to you right from the factory (or an authorized dealer).  


  New engines have a "zero-time" logbook and no operating history because they've never been installed on an aircraft. Usually there's a "test cell run" entry in the logbook, if anything.

Rebuilt (There's not a common abbreviation here - I've seen SFRB or SFRM)
An engine that went back to the factory and was rebuilt to the same production limits as a new engine. Some of the parts in the engine will be new, some may be used, but all will meet the same standards and specifications  as a new engine.  


  Only the original manufacturer can rebuild an engine, and it comes to you with a zero time logbook and no operating history (because a rebuilt engine is done by the factory and is to new-engine limits and may be made up of components from a bunch of other engines there's no way to give it a history, so it's treated the same way new engines are).

Overhauled (SMOH or SFOH)
An engine that has been completely disassembled (the crank case is split and all the parts pulled out), extensively inspected, and reassembled to either service limits or new limits. An overhaul usually includes some used parts (frequently from the same engine), and new parts as necessary to meet limits.
An overhaul can be done by the factory ("SFOH" - Since Factory Overhaul), or by a specialty engine shop, or even by your local A&amp;P mechanic (both "SMOH" - Since Major Overhaul).  


  Overhauled engines retain their operating history, and should have a complete set of logbooks since the day the engine was manufactured. Even if the only part that gets reused is the engine case an overhaul is still "the same engine" in the FAA's eyes, so make sure you can trace the full operating history of that engine.



You will sometimes see another term - "since top overhaul" (STOP or STOH), but this one is not actually defined in the regulations. Generally it means a shop removed and replaced all the cylinders (the "top end"), but did not open the crank case and thus may not have inspected the "bottom end" (crankshaft, camshaft, bearings, etc.)



The terms "overhaul" and "rebuild" are defined in 14 CFR 43.2, and further expounded upon in AC 20-62E: 


  §43.2   Records of overhaul and rebuilding. (a) No person may describe
  in any required maintenance entry or form an aircraft, airframe,
  aircraft engine, propeller, appliance, or component part as being
  overhauled unless—
  
  (1) Using methods, techniques, and practices acceptable to the
  Administrator, it has been disassembled, cleaned, inspected, repaired
  as necessary, and reassembled; and
  
  (2) It has been tested in accordance with approved standards and
  technical data, or in accordance with current standards and technical
  data acceptable to the Administrator, which have been developed and
  documented by the holder of the type certificate, supplemental type
  certificate, or a material, part, process, or appliance approval under
  part 21 of this chapter.
  
  (b) No person may describe in any required maintenance entry or form
  an aircraft, airframe, aircraft engine, propeller, appliance, or
  component part as being rebuilt unless it has been disassembled,
  cleaned, inspected, repaired as necessary, reassembled, and tested to
  the same tolerances and limits as a new item, using either new parts
  or used parts that either conform to new part tolerances and limits or
  to approved oversized or undersized dimensions.




There is no difference between a "factory" overhaul and someone else's "major" overhaul - at least according to the regulations: The standards for what constitutes an overhaul are the same.  What is different is who you talk to if the engine breaks (for a factory overhaul you would go back to the factory, for an overhaul by an engine shop you would go to the shop, and for one by your local mechanic you would go talk to the mechanic).  

If you're looking at a plane which has had an engine overhaul (the vast majority of the used piston GA fleet) you should pay attention to where it was done and the reputation of the overhauler, but given two engines, one overhauled at the Lycoming factory and the other at a reputable shop like Penn Yan Aero, they would generally be considered equivalent.