What is an Aircraft Overhaul?
Briefly, "overhaul" isn't a strictly defined term but it's generally used to refer to the intensive inspection, maintenance and repair of an aircraft component, frequently - but not exclusively - the engines.

The word overhaul by itself is a general term that's used in all sorts of places but it's not defined in 14 CFR 1.1, for example. The most specific definition I could find is in this agreement between the FAA and EASA:


  "Overhaul" means a process that ensures the aeronautical article is
  in complete conformity  with the applicable service tolerances
  specified  in the type certificate  holder's, or equipment 
  manufacturer's instructions for  continued airworthiness, or in  the
  data which is approved or  accepted by the Authority.
  
  No person may
  describe an article as being overhauled unless it  has been at least 
  disassembled, cleaned, inspected, repaired as necessary, reassembled
  and tested in accordance  with the above-specified data.


Note that that definition only mentions "aeronautical articles", which is very broad indeed and not limited to just engines, airframes or anything else.

In the general aviation world, a very common usage of "overhaul" is in the context of piston engines, which have a Time Between Overhaul (TBO or TBOH) set by the manufacturer. Once an engine reaches that number of hours, it needs to be overhauled. But even there, the term isn't precise and the FAA has a circular called Reciprocating Engine Overhaul Terminology and Standards that explains what it usually means:


  c. Overhaul.  In the general aviation industry, the term engine
  overhaul has two  identifications that make a  distinction between the
  degrees of work on an engine:
  
  (1) Major Overhaul.  A major overhaul
  consists of the complete disassembly of an  engine. The overhaul
  facility inspects the engine, repairs it as necessary, reassembles,
  tests, and  approves it for return to service within the  fits and
  limits specified by the manufacturer’s  overhaul data. This could be
  to new fits and limits or serviceable limits. The engine owner  should
  clearly understand what fits and limits should be used when  the
  engine is presented for  overhaul. The owner should also be aware of
  any replaced parts, regardless of condition, as a  result of a
  manufacturer’s overhaul data,  SB, or an Airworthiness Directive (AD).
  
  (2) Top Overhaul.  Top overhaul consists of repair to  parts outside
  of the crankcase, and  can be accomplished without completely
  disassembling the entire engine. It can include the  removal of
  cylinders, inspection and repair to cylinders, inspection and repair
  to cylinder walls,  pistons, valve-operation mechanisms, valve guides,
  valve seats, and the replacement of piston  and piston rings. All
  manufacturers do not recommend a top overhaul. Some manufacturers 
  indicate that a powerplant requiring work to this extent  should
  receive a complete overhaul.


Apart from that usage, the term MRO (Maintenance, Repair and Overhaul) seems to be used in the airline industry to describe maintenance facilities, but there isn't any specific definition that I could find.

So there's no single definition of "overhaul". It just means that you've very thoroughly inspected and tested something on an aircraft, whether it's an engine or a wheel, and you've verified that it's fit to be used in flight, i.e. it's airworthy. The reasons why it's important are probably obvious; they're the same reasons that your car needs regular servicing to run safely, efficiently and reliably.