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.