These quantities can have a geometric meaning and are also useful in matrix algebra, the geometric meaning is discussed on this page, it tells us something about the symmetry of a transform.
An eigenvector is a vector whose direction is not changed by the transform, it may be streached, but it still points in the same direction.
Each eigenvector has a corresponding eigenvalue which gives the scaling factor by which the transform scales the eigenvector. So the eigenvector is a vector and the eigenvalue is a scaler.
A given transform may have more than one eigenvector and eigenvalue pair depending on how many dimensions we are working in. For instance:
 If we are working in 2 dimensions there are upto 2 eigenvector and eigenvalue pairs.
 If we are working in 3 dimensions there are upto 3 eigenvector and eigenvalue pairs.
and so on.
As an example, if we have a rotation transform in 3 dimensions, then the eigenvector would be the axis of rotation since this is not altered by the transform and the corresponding eigenvalue would be +1 since the axis is not scaled by the rotation. If we have a rotation in 2 dimensions then the eigenvectors would be ±i where i is √1 since all vectors in the plane change direction.
Eigenvalues
The eigenvalues of a matrix [M] are the values of λ such that:
[M] v = λ v
where:
 v = eigenvector
 λ = lambda = eigenvalue
this gives:
M  λ I = 0
where I = identity matrix
this gives:

= 0 
so
(m00 λ) (m11 λ) (m22 λ) + m01 m12 m20 + m02 m10 m21  (m00 λ) m12 m21  m01 m10 (m22 λ)  m02 (m11 λ) m20 = 0
the values of λ are the eigenvalues of the matrix
Eigenvectors
Associated with each eigenvalue λ_{i} is an eigenvector {u_{i}} such that:
[M] {u_{i}} = λ_{i} {u_{i}}
where:
 [M] is a matrix
 λ_{i} is its eigenvalues (i=1,2,3)
 {u_{i}}is its eigenvectors
Program
There are a number of open source programs that can calculate eigenvalues and eigenvectors. I have used Axiom, how to install Axiom here.
To get a numeric solution for a given matrix, we can use eigenvalues(m) and eigenvectors(m) as shown here:
I have put user input in red:
(1) > m := matrix[[1,4,7],[2,5,8],[3,6,9]]
+1 4 7+   (1) 2 5 8   +3 6 9+ Type: Matrix Integer (2) > ev := eigenvalues(m)
2 (2) [0,%A  %A  15%A  18] Type: List Union(Fraction Polynomial Integer,SuchThat(Symbol, Polynomial Integer)) (3) > eigenvectors(m)
(3) + 1 +   [[eigval= 0,eigmult= 1,eigvec= [ 2]],   + 1 + +%G  12+   6  2   [eigval= (%G  %G  15%G  18),eigmult= 1,eigvec= [%G  6 ]]]    12    + 1 + Type: List Record(eigval: Union(Fraction Polynomial Integer,SuchThat(Symbol,Polynomial Integer)),eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer) 
Or we can find a general formula for a given matrix as shown here:
(1) > msymb := matrix[[a,b,c],[d,e,f],[g,h,i]] +a b c+   (1) d e f   +g h i+ Type: Matrix Polynomial Integer (2) > evsymb := eigenvalues(msymb) (2) [ %B  2 ((a  %B)e  b d  %B a + %B )i + (( a + %B)f + c d)h + 2 2 3 (b f  c e + %B c)g + ( %B a + %B )e + %B b d + %B a  %B ] Type: List Union(Fraction Polynomial Integer, SuchThat(Symbol,Polynomial Integer)) (3) > eigenvectors(msymb) (3) [ [ eigval = %H  2 ((a  %H)e  b d  %H a + %H )i + (( a + %H)f + c d)h + 2 2 3 (b f  c e + %H c)g + ( %H a + %H )e + %H b d + %H a  %H , eigmult= 1, + 2 2 +  ((e  %H)h + b g)i  f h + ( c g  %H e + %H )h  %H b g      2 2   d h + ( e + a)g h  b g    eigvec= [ 2 2 ]] ( d h + ( a + %H)g)i + (f g + %H d)h + c g + (%H a  %H )g   2 2   d h + ( e + a)g h  b g    + 1 + ] Type: List Record(eigval: Union(Fraction Polynomial Integer, SuchThat(Symbol,Polynomial Integer)) ,eigmult: NonNegativeInteger,eigvec: List Matrix Fraction Polynomial Integer) 