What is the doctrine of the Trinity? What is the doctrine of the Trinity as defined by Christianity? 

Obviously there are three entities involved: God the Father, Christ, and the Holy Spirit, but is there more to the definition?
As you already understand, the doctrine of the Trinity states that there is one God, who exists in three persons. The Father, the Son, and the Holy Spirit.  

From http://carm.org/trinity


  The Father is not the same person as the Son; the Son is not the same
  person as the Holy Spirit; and the Holy Spirit is not the same person
  as Father.  They are not three gods and not three beings.  They are
  three distinct persons; yet, they are all the one God.  Each has a
  will, can speak, can love, etc., and these are demonstrations of
  personhood.  They are in absolute perfect harmony consisting of one
  substance. They are coeternal, coequal, and copowerful.  If any one of
  the three were removed, there would be no God.


...


  There is, though, an apparent separation of some functions among the
  members of the Godhead.  For example, the Father chooses who will be
  saved (Eph. 1:4); the Son redeems them (Eph. 1:7); and the Holy Spirit
  seals them, (Eph. 1:13).


I'd like to expand on that, but there's not much I could expand on that hasn't already been covered in depth on this site.  And while it's really not a complex doctrine, all sorts of heretical teachings come about based on a misunderstanding of it, or a rejection of it.   Rather than going into any more detail here, I'd just recommend browsing the various questions tagged "Trinity" here.