Basic Definition :

   SS(sum of squares) : the sum of squares of any list of n real numbers, a_1, ..., a_n,  refers to
               (a_1 - \bar a )^2 + ....   + (a_n - \bar a)^2
          where \bar a = average of a_i
   [Note ; if  \bar a =0, then SS is nothing but the squared length of the vector  a = (a_1, ...a_n)' ]
   [ Geometrically,  the vector (\bar a, ...\bar a)' is the projection of  vector a  in  the direction of (1, .., 1)]
   [ The vector (a_1-\bar a, ...a_n-\bar a)' is the orthogonal complement ]
   [  From Pythagorean thorem , we can see  why  ||a||^2 = SS + n (\bar a)^2  must hold. ]
 

One-way ANOVA :
     The data  are grouped (classified, partitioned ) into several groups (classes, clusters) :
      Group 1 :  (Y_11, Y_12, .... Y_1n_1)
     Group 2 : (Y_21, ....     , Y_2n_2)
    ...............
     Group k  : (Y_k1, ... Y_k n_k)

 In total we have n = n_1 + ... + n_k obervations.

 Total Sum of Squares :    Apply SS to the list of  n  Y values
  Between group Sum of squares :
            (1) create a list of n values by replacing each Y_ij with the corresponding group mean; call this group-mean vector
                          (remember that it is also an n-dimensional  vector, not a k-dimensional vector)
              (2) Apply SS to this new list of n values.
Within group Sum of squares (also called error sum of squares, residual sum of squares) :
             (1) Create a list of n values by replacing each Y_ij with the difference between Y_ij and its  group mean
             (2) Apply Sum of squares to (1)

 Again the ANOVA identity holds because of Pythagorean theorem.
There are three spaces :

(1)  S_1 : The dimensional space generated by the vector of ones,  (1, 1, ..1)'.
(2)  S_2 : The k dimensional space generated by k  group indicator vectors, e_1, ..e_k. For instance,
          e_2 = (0,....0, 1, 1,....,1, 0,0,.....) where the places for 1  corresponding to the second group.
(3)  S_3 : The entire n dimensional space.
[Note : S_1 is within S_2 and S_2 is within S_3. Thus these spaces are nested.]

The projection of Y vector onto space S_1  is the vector of grand mean.
The projection of Y onto  S_2 is the vector of group means as  created in the first step of Between
The projection of group-mean vector  onto S_1 =   vector of grand mean.

Now to see how the Pythogrean Theorem  comes in,  we need a 3-D diagram ,

vector C = vector Y - vector of grand mean
vector A = group-mean vector   - vector of grand mean
vector  B = vector Y -  group-mean vector
Geometry shows that A, B are othoognal to each other, while algeraically C=A+B ; so Pythogorean applies :
 
||C||^2 = ||A||^2 + ||B||^2
from which  we can easily see the ANOVA identity holds because

||C||^2 = total sum of squares
||A||^2 = Between group sum of squares
||B||^2 = Within group sum of squares.