A (mathematically) pure function is a function that always returns the same output if called with the same parameters. Their return values depend on their parameters and their parameters only. Additionally, pure functions have no side-effects.
To put it differently, calling them does not affect the program's environment and their results do not change if their parameters don't change. They don't access the filesystem, they don't send out network packets, nor do they output CLI messages.
Here's an example of a pure function in Python:
def hypotenuse(a, b): return math.sqrt(a**2 + b**2)
But don't let this fool you; even much more complex, powerful functions can be pure.
Impure functions, among other things, include functions performing input/output operations, randomness generators, functions spawning threads or forking -- all non-deterministic functions.
While pure functions are mostly considered in the context of functional programming languages, this does not mean they aren't useful in languages that are more "traditional".
The majority of widely-used programming languages do not recognize the concept of a pure function. That doesn't mean pure functions are less useful in such languages. They are still extremely important for ensuring the correctness and maintainability of the codebase.
It is obvious we should strive to make our functions pure, whenever possible.
Although practicality beats purity.
No need to be dogmatic about it, though. Sometimes it just doesn't work.
Haskell and similar languages naturally push us to structure our code in a way that separates side effects from pure computation. In conventional languages, this must be practiced more deliberately.
A good piece of advice, I often find myself returning to, is to keep all operations with side-effects as close to the entry point of the program as possible. You would, for instance, open the config file in the main part of the program (or close to it) and pass its contents to a pure function that would parse it, rather than opening the file somewhere deep in the call stack.
Programs that achieve this are more transparent -- it is more clear in what ways the program interacts with its environment.
Try to put the majority of complex program logic into pure functions while keeping impure functions as simple as possible.
This will greatly simplify testing, as it will maximize the amount of code you can cover with simple unit tests and reduce the number of required integration tests.
It is generally accepted test-driven development can be very beneficial to the practice of software development.
While I am not convinced tests should always be written before the actual code, I find it very important to at least think about how the code will be tested before writing it. This tends to drive me towards having more pure functions.
In languages, that don't have a type system that can differentiate between pure and impure functions, some degree of effort is required to ensure the functions are actually pure.
There can be no calls to non-pure functions, the code must not access immutable global variables, the functions should not have any internal state (think generators in Python), etc.
In certain programming languages, it can be hard to be 100% sure about the purity of a function. Some languages even perform impure operations, such as heap allocations, behind your back.
It is, however, still better to have an almost-pure function than a blatantly impure one.