HC in general means OverHead Cam while SOHC means Single OverHead Cam.
In the SOHC engine the camshaft is installed in the cylinder head and valves are operated either by the rocker arms or directly through the lifters (as in the picture).
The advantage is that valves are operated almost directly by the camshaft - easy to achieve the perfect timing at high rpm. Also it's possible to install three or four valves per cylinder
The disadvantage - an OHC engine requires a timing belt or chain with related components - more complex and more expensive design.
DOHC or Twin cam engine
DOHC or Double OverHead Cam - the setup used in many today's cars. Since it's possible to install multiple valves per cylinder and place intake valves on the opposite side from exhaust vales, DOHC engine can "breath" better meaning that it can produce more horsepowers with smaller engine volume. Compare: The 3.5-liter V6 DOHC engine of 2003 Nissan Pathfinder has 240 hp, similar to 245 hp of the 5.9-liter V8 OHV engine of 2003 Dodge Durango.
Pros: High efficiency, possible to install multiple valves per cylinder and adopt variable timing.
Cons: More complex and more expensive design.