You don't specify a version, and file objects behave a little bit differently between Python 2 and Python 3. The general idea is the same, but some of the specific details are different. The following assumes you're using Python 3, or that you're using the version of open from the io module in Python 2.6 or 2.7 rather than Python 2's builtin open.
It isn't a file pointer, although there's a good chance it is implemented in terms of one behind the scenes. Unlike C, Python does not expose the concept of pointers.
However, what you seem to be thinking of is the 'stream position', which is kindof similar to a pointer. This is the number reported by tell(), and which can be fed into seek(). For binary files, it is a byte offset from the start of the file. In text files, it is just 'an offset' which is meaningful to the file object - the docs call it an "opaque number" (ie, it has no defined physical meaning in terms of how the file is stored on disk). But in both cases, it is an offset from the start, and therefore the start is zero. This is only true if the file supports random access - which you usually will be, but be prepared to eventually run into a situation where you're not - in which case, seek and tell raise errors.
Like the instruction pointer in processors, the stream position is where the next operation will start from, rather than where the current one finished. So, yes, after you've written a string to the file, the current position will usually be one offset value past that.
When you've just opened a file, the offset will usually be zero or the end of the file (one higher than the maximum value you could read from without getting EOF). It will be zero if you've opened it in 'r' mode, the end if you've opened it in 'a' mode and the two are equivalent for 'w' and 'w+' modes since those truncate the file to zero bytes.
