Transfiguration is magic which changes one object into another. A living thing can be changed into a non - living thing and a non - living thing can be changed into a living thing.  Professor 
Mc Gonagall changed her desk into a pig and then back again to demonstrate the possibilities of transfiguration to her first year students.  It is also possible to change one part of a living thing. The Transfiguration class at hogwarts is taught by Professor McGonagall.  Dumbledore's speciality was also transfiguration.  He was the transfiguration teacher fifty years before this time.

 "Transfiguration is some of the most complex and dangerous magic you will learn at Hogwarts. Anyone messing around in my class will leave and not come back. You have been warned."
Minerva McGonagall

Transfiguration was a difficult class involving very complicated notes and calculations.  Professor Mc Gonagall gave a great deal of homework.  class work for beginners included trying to change a match into a needle.  By the third year exams, students were expected to be able to turn a teapot into a tortoise.

A fourth year student should be mastering crosspieces itches, such as changing a guinea fowl into guinea pig, changing a porcupine into pincushion,  and changing buttons into beetles. 
During the sixth year at hogwarts, human transfiguration is taught.

students sometimes use this ability either accidently or on purpose.  a student accidentally changes his friend into a badger in class.  Hermione's teeth were turned into beaver teeth by malfoy when he was trying to put a spell on harry.

The textbooks for these classes are A Beginner's Guide to Transfiguration, Intermediate Transfiguration, Advanced Transfiguration.  they can be purchased at flourish and blotts in diagon alley.

somewhat related to transfiguration is animagus.  animagus is the ability of a witch or wizard to transform oneself into an animal while retaining his or her magical powers.  the wizard's or witches personality determine what animal form they will assume.  this is a rather rare ability and takes years of practice to perfect.  The ministry of magic keeps a discription  of the animal the wizard or witch assumes. Unfortunately not all animagus register with the

mentioned animagi

Minerva McGonagall = cat.
James Potter = stag (prongs)
sirius black = black dog (padfoot)
peter pettirew = rat (wormtail)
rita skeeter = beetle