1. External libraries
A very complete list can be found at PyPi the Python Package Index: To install, use pip, which comes with Python: pip install package or download, unzip, and run the installer directly from the directory: python setup.py install If you have Python 2 and Python 3 installed, use pip3 (though not with Anaconda) or make sure the right version is first in your PATH.

2. Numpy
Mathematics and statistics, especially multi-dimensional array manipulation for data processing. Good introductory tutorials by Software Carpentry:

3. Numpy data
Perhaps the nicest thing about numpy is its handling of complicated 2D datasets. It has its own array types which overload the indexing operators. Note the difference in the below from the standard [1d][2d] notation:
import numpy
data = numpy.int_([
[1,2,3,4,5],
[10,20,30,40,50],
[100,200,300,400,500] ])
print(data[0,0]) # 1
print(data[1:3,1:3]) # [[20 30][ ]]
On a standard list, data[1:3][1:3] wouldn't work, at best data[1:3][0][1:3] would give you [20][30]

4. Numpy operations
You can additionally do maths on the arrays, including matrix manipulation.
import numpy
data = numpy.int_([
[1,2,3,4,5],
[10,20,30,40,50],
[100,200,300,400,500] ])
print(data[1:3,1:3] - 10) # [[10 20],[ ]]
print(numpy.transpose(data[1:3,1:3])) # [[20 200],[30 300]]
There's a nice numpy cheatsheet from datacamp at:

5. Pandas
Data analysis. Based on Numpy, but adds more sophistication.

6. Pandas data
Pandas data focuses around DataFrames, 2D arrays with addition abilities to
name and use rows and columns.
import pandas
df = pandas.DataFrame(
data, # numpy array from before.
index=['i','ii','iii'],
columns=['A','B','C','D','E'] )
print (data['A'])
print(df.mean(0)['A'])
print(df.mean(1)['i'])
Prints:
i 1
ii 10
iii 100
Name: A, dtype: int32
37.0
3.0

7. scikit-learn
Scientific analysis and machine learning. Used for machine learning. Founded on Numpy data formats.

8. Beautiful Soup
Web analysis. Need other packages to actually download pages like the library requests. BeautifulSoup navigates the Document Object Model: Not a library, but a nice intro to web programming with Python.

9. Tweepy
Downloading Tweets for analysis. You'll also need a developer key: access-key-for-twitter-oauth/994/ Most social media sites have equivalent APIs (functions to access them) and modules to use those.

10. NLTK
Natural Language Toolkit. Parse text and analyse everything from Parts Of Speech to positivity or negativity of statements (sentiment analysis).

11. Celery
Concurrent computing / parallelisation. For splitting up programs and running them on multiple computers e.g. to remove memory limits. See also: