While not essential for the operation of a pressure washer, quick connectors make life a lot easier when setting up or packing up after a hard days work!  There are a variety of quick connectors available for the pressure washer.   Also known as quick couplers or quick disconnect.

GARDEN HOSE CONNECTORS
For supplying water to a pressure washer through a garden hose there are many different brands of quick connectors available. 

For the homeowner a cheap plastic style, available from most hardware or plumbing stores will do the job.  

For the commercial operator a more robust, and expensive,  brass connector is preferred.  

The threads on a garden hose are known as GHT (garden hose thread).  A female GHT is known as FGHT (usually the end that fits on to the tap).  A male GHT is known as MGHT (usually the end that fits on to the pressure washer). So when using a 2 piece quick connector (QC) the FGHT side fits on to the tap and the MGHT fits on to the hose. When using a QC on the pressure washer the reverse is normal.  However, some pressure washer manufacturers use a MGHT on the inlet to the pressure washer.  In this case the manufacturer normally supplies a QC with both parts being FGHT.  

A FGHT X FGHT QC can sometimes be difficult to find.

HIGH PRESSURE HOSE CONNECTORS
High Pressure QC'c are available in a few styles.  

The most common North American style is the 2 piece socket and plug.
The threaded sides of these connectors are typically NPT (National Pipe Thread), either Male or Female.  Common thread sizes are 1/4 inch and 3/8 inch.  !/4 inch size is typically used on the end of the wand for nozzles and 3/8 inch are used on high pressure hoses.  The socket side has a few moving parts which allow for the plug side to easily fit, as well as lock the plug into place.  The key to a good seal under pressure is the o-ring, which prevents water from leaking out using the pressure against it.  The plug side is one piece, which fits into the coupler and held is tight to the o-ring.  Here are some of the socket and coupler types available:
  Female Socket            Male Socket
Female Plug                  Male Plug

EZGrip Female Socket      EZGrip Male Socket

An o-ring fits inside the Socket to make the seal

The other style of QC is the Screw Type (most commonly referred to as M22 because the screw threads are metric 22 millimeter)  Mostly a European thing these connectors are quite prevalent in North America.  While not as easy to use they are very secure.  They work by screwing the socket and plug sides together with about 4 turns.  The socket side has a large plastic cover which allows it to be turned easily by hand.  If you drop the socket on concrete this plastic cover will break, making the fitting pretty much useless as a quick connector.  This style comes in both male and female NPT threads to fit wands and hoses.  The socket side has an o-ring to make a positive seal. Here are some photos of typical M22 fittings:

Female M22 Plug   Male M22 Plug

    Female M22 Socket  Male M22 Socket

Female X Female M22 Socket Connector

M22 Socket X QC Plug Adapter

NOTE ADDED 01/22/2017:  There are two different internal diameter sizes available on M22 fittings.  Some are 15mm, while others are 14mm.  This can cause problems when replacing hoses and guns on small "hobby" electric pressure washers.  There are adapters available for 15mm to 14mm and 14mm to 15mm.  See:
https://www.pressurewashersonline.ca/products/twist-seal-coupler-m22-adapters
Thanks to David Thompson for pointing this omission out.

Regardless of what QC you use make sure you keep them clean.  Dragging your hoses around a dirty or muddy site is a sure fire way to fill the connectors with debris. Connecting dirty connectors will contaminate your system, plugging nozzles, breaking valves, and scoring pistons.

Another thing to be aware of is using QC style fitting when working over the side of a building.  This fitting can be pulled apart when pulled over an edge, allowing hoses to fall.  M22 screw types will not come apart when pulled.