The transporters are a very complicated system because everything must be checked over and over to ensure a safe transport.  No one wants to lose an arm after transport so many precautions are taken.

     There are a lot of pieces.  I'm not going to talk about all of them because the specifics are very boring.  But, here is an overview of what it can and cannot do.
     Transporters can be set to 2 different types of resolution.  Quantum resolution is for life forms. There are 6 transporters that use the quantum resolution under normal conditions.  The other is molecular resolution, which are used for cargo and other non-living things.  There are 8 transporters that use the molecular resolution under normal conditions.  But, any transporter can be set to use molecular or quantum resolution.  In addition to those 14 transporters, there are 6 more emergency transporters throughout the ship.  These can only be used to beam from the ship and have a limited range.  They are also special because they can operate under reduced power.
     There are 4 basic stages to the transport cycle.  During the first stage, the coordinates are entered into the computer, the location is checked for environment and range, and finally diagnostic procedures make sure the transporter is working.  It would be a shame if we lost some crewmen in transport.  During the next stage, a "photograph" of the subject is taken.  This "photograph" tells where all the pieces of the matter stream will go when the subject reaches its destination.  Then the subject is broken up into subatomic matter.  Sounds like it hurts, don't it?  During the third stage, the matter stream is held in a pattern buffer to compensate for the Doppler shift .  Also, the transport can be aborted to another chamber if something goes wrong.  During the last stage, the matter stream is transmitted to the destination from of the emitter pad arrays.
     In addition to beaming up and beaming down objects, the transporters can be designed to do a number of other things.  One use is to disengage the annular confinement beam (ACB).  The ACB does the materializing and dematerializing.  When it is deactivated the subject will form at random.  Usually they look like random particles and gases.  They call this dispersal.  It is used to disperse dangerous transport subjects such as explosives.  They molecules are then sent into space.  Another use is the biofilter scan.  When a subject is in the transporter it is scanned for traces of bad bacteria and viruses.  Then the virus/bacteria can be removed in the transporter before the subject materializes.
     Although the transporter has such great possibilities, it also has many limitations.  One such limitation is the range, which is only 40,000 km and that is rare due to energy availability.  Another is the duty cycle.  The pattern buffer, the place where the matter stream is held for Doppler compensation, has a cooldown period of 87 seconds.  There are 3 pattern buffers on board but after 2 transports, because 2 have to be active in case of emergency, one must wait for it to reset.  This severely limits the transporters.  Here's a pattern buffer.

     Look at the blue stream swirling around inside.  That could be you in the future.  Scary isn't it?