Can you tell me when to use these vectorization methods with basic examples?

I see that map is a Series method whereas the rest are DataFrame methods. I got confused about apply and applymap methods though. Why do we have two methods for applying a function to a DataFrame? Again, simple examples which illustrate the usage would be great!

apply works on a row / column basis of a DataFrame
applymap works element-wise on a DataFrame
map works element-wise on a Series

Straight from Wes McKinney's Python for Data Analysis book, pg. 132 (I highly recommended this book):

Another frequent operation is applying a function on 1D arrays to each column or row. DataFrame’s apply method does exactly this:

In [116]: frame = DataFrame(np.random.randn(4, 3), columns=list('bde'), index=['Utah', 'Ohio', 'Texas', 'Oregon'])
In [117]: frame
Out[117]:
b         d         e
Utah   -0.029638  1.081563  1.280300
Ohio    0.647747  0.831136 -1.549481
Texas   0.513416 -0.884417  0.195343
Oregon -0.485454 -0.477388 -0.309548
In [118]: f = lambda x: x.max() - x.min()
In [119]: frame.apply(f)
Out[119]:
b    1.133201
d    1.965980
e    2.829781
dtype: float64

Many of the most common array statistics (like sum and mean) are DataFrame methods, so using apply is not necessary.

Element-wise Python functions can be used, too. Suppose you wanted to compute a formatted string from each floating point value in frame. You can do this with applymap:

In [120]: format = lambda x: '%.2f' % x
In [121]: frame.applymap(format)
Out[121]:
b      d      e
Utah    -0.03   1.08   1.28
Ohio     0.65   0.83  -1.55
Texas    0.51  -0.88   0.20
Oregon  -0.49  -0.48  -0.31

The reason for the name applymap is that Series has a map method for applying an element-wise function:

In [122]: frame['e'].map(format)
Out[122]: 
Utah       1.28
Ohio      -1.55
Texas      0.20
Oregon    -0.31
Name: e, dtype: object