Why does being thread safe matter in a web app? Pylons (Python web framework) uses a global application variable which is not thread safe. Does this matter? Is it only a problem if I intend on using multi-threading? Or, does it mean that one user might not have updated state if another user... I'm just confusing myself. What's so important with this?
Threading errors can lead to serious and subtle problems.
Say your system has 10 members. One more user signs up to your system and the application adds him to the roster and increments the count of members; "simultaneously", another user quits and the application removes him from the roster and decrements the count of members.
If you don't handling threading properly, your member count (which should be 10) could easily be nine, 10, or 11, and you'll never be able to reproduce the bug.
So be careful.
You should care about thread safety. E.g in java you write a servlet that provides some functionality. The container will deploy an instance of your servlet, and as HTTP requests arrive from clients, over different TCP connections, each request is handled by a separate thread which in turn will call your servlet. As a result, you will have your servlet being call from multiple threads. So if it is not thread-safe, then erroneous result will be returned to the user, due to data corruption of access to shared data by threads.
It really depends on the application framework (which I know nothing about in this case) and how the web server handles it. Obviously, any good webserver is going to be responding to multiple requests simultaneously, so it will be operating with multiple threads. That web server may dispatch to a single instance of your application code for all of these requests, or it may spawn multiple instances of your web application and never use a given instance concurrently.
Even if the app server does use separate instances, your application will probably have some shared state--say, a database with a list of users. In that case, you need to make sure that state can be accessed safely from multiple threads/instances of your web app.
Then, of course, there is the case where you use threading explicitly in your application. In that case, the answer is obvious.
Your Web Application is almost always multithreading. Even though you might not use threads explicitly. So, to answer your questions: it's very important.
How can this happen? Usually, Apache (or IIS) will serve several request simultaneously, calling multiple times from multiple threads your python programs. So you need to consider that your programs run in multiple threads concurrently and act accordingly.
(This was too long to add a comment to the other fine answers.)
Concurrency problems (read: multiple access to shared state) is a super-set of threading problems. The (concurrency problems) can easily exist at an "above thread" level such as a process/server level (the global variable in the case you mention above is process-unique value, which in turn can lead to an inconsistent view/state if there are multiple processes).
Care must be taken to analyze the data consistency requirements and then implement the software to fulfill those requirements. I would always err on the side of safe, and only degrade in carefully analyzed areas where it is acceptable.
However, note that CPython runs only one thread context for Python code execution (to get true concurrent threads you need to write/use C extensions), so, while you can get a form of race condition upon expected data, you won't get (all) the same kind of partial-write scenarios and such that may plague C/C++ programs. But, once again. Err on the side of a consistent view.
There are a number of various existing methods of making access to a global atomic -- across threads or processes. Use them.