When to use NSInteger vs. int

You usually want to use NSInteger when you don't know what kind of processor architecture your code might run on, so you may for some reason want the largest possible integer type, which on 32 bit systems is just an int, while on a 64-bit system it's a long.

I'd stick with using NSInteger instead of int/long unless you specifically require them.

NSInteger/NSUInteger are defined as *dynamic typedef*s to one of these types, and they are defined like this:

#if __LP64__ || TARGET_OS_EMBEDDED || TARGET_OS_IPHONE || TARGET_OS_WIN32 || NS_BUILD_32_LIKE_64
typedef long NSInteger;
typedef unsigned long NSUInteger;
#else
typedef int NSInteger;
typedef unsigned int NSUInteger;
#endif

With regard to the correct format specifier you should use for each of these types, see the String Programming Guide's section on Platform Dependencies

Why use int at all?

Apple uses int because for a loop control variable (which is only used to control the loop iterations) int datatype is fine, both in datatype size and in the values it can hold for your loop. No need for platform dependent datatype here. For a loop control variable even a 16-bit int will do most of the time.

Apple uses NSInteger for a function return value or for a function argument because in this case datatype [size] matters, because what you are doing with a function is communicating/passing data with other programs or with other pieces of code; see the answer to When should I be using NSInteger vs int? in your question itself...

they [Apple] use NSInteger (or NSUInteger) when passing a value as an argument to a function or returning a value from a function.