Exploiting the features of the Python language to produce code that is clear, concise and maintainable.
Pythonic means code that doesn't just get the syntax right but that follows the conventions of the Python community and uses the language in the way it is intended to be used.
This is maybe easiest to explain by negative examples, as in the linked article from the other answers. Examples of unpythonic code often come from users of other languages, who instead of learning a Python programming patterns such as list comprehensions or generator expressions, attempt to crowbar in patterns more commonly used in C or java. Loops are particularly common examples of this.
For example in Java I might use
for i in (i; i < items.length ; i++)
n = items[i];
... now do something
In Python, we can try and replicate this using while loops but it would be cleaner to use
for i in items:
Or, even a generator expression
(i.some_attribute for i in items)
So essentially when someone says something is unpythonic, they are saying that the code could be re-written in a way that is a better fit for pythons coding style.
Typing import this at the command line gives a summary of Python principles. Less well known is that the source code for import this is decidedly, and by design, unpythonic! Take a look at it for an example of what not to do.