Automatically backup website files and database
The names are chosen well. The code is clear. Obviously it is for a specialized case so there are many constants in the code - that is OK. It is strictly procedural, which is a good thing - OOP has a tendency to just become boilerplate for small things like these (it would be useful if you intended to make a framework for a general backup tool)

I would definitely remove the unnecessary variables.


db_passwords = {'somedatabasename': 'somedbpassword'} which is only used a few lines later as db_passwords[db_name], take the password as an argument to the function instead.
date = time.strftime('%Y-%m-%d'), it would be understandable if you had to use it twice and be sure it still was the same day (in that case I would argue for a function argument), but in your case you only use it once. So remove the variable and do instead ... 'date': time.strftime('%Y-%m-%d').


Some variables could be better as functions, for example
 - zip_command, I would put that into a function which would take app_name and filename as argument and then return the command. Of course, the day you want different zip commands depending on database size or backup time you just add an extra argument.

Perhaps more subjective but I would go for the new Advanced String Formatter. 

To generalize your tool, I would do one of the two: a) put your options on the command line (e.g. argparse or optparse) or b) use a configuration file (fabric can take an environment file as argument). I would absolutely only put those options that do vary between the sites (rather than making everything possible with an explosion in code for no added real utility).