One source of difficulty with this question is that you have a program named bar/bar.py: import bar imports either bar/__init__.py or bar/bar.py, depending on where it is done, which makes it a little cumbersome to track which a is bar.a.
Here is how it works:
The key to understanding what happens is to realize that in your __init__.py,
from bar import a
in effect does something like
a = bar.a
# … where bar = bar/bar.py (as if bar were imported locally from __init__.py)
and defines a new variable (bar/__init__.py:a, if you wish). Thus, your from bar import a in __init__.py binds name bar/__init__.py:a to the original bar.py:a object (None). This is why you can do from bar import a as a2 in __init__.py: in this case, it is clear that you have both bar/bar.py:a and a distinct variable name bar/__init__.py:a2 (in your case, the names of the two variables just happen to both be a, but they still live in different namespaces: in __init__.py, they are bar.a and a).
Now, when you do
import bar
print bar.a
you are accessing variable bar/__init__.py:a (since import bar imports your bar/__init__.py). This is the variable you modify (to 1). You are not touching the contents of variable bar/bar.py:a. So when you subsequently do
bar.foobar()
you call bar/bar.py:foobar(), which accesses variable a from bar/bar.py, which is still None (when foobar() is defined, it binds variable names once and for all, so the a in bar.py is bar.py:a, not any other a variable defined in another module—as there might be many a variables in all the imported modules). Hence the last None output.
Conclusion: it is best to avoid any ambiguity in import bar, by not having any bar/bar.py module (since bar.__init__.py makes directory bar/ a package already, that you can also import with import bar).