With the introduction of Swift I\'ve been trying to get my head round the new language
I\'m an iOS developer and would use types such as NSString, NSInteger, N
You should use the Swift native types whenever possible. The language is optimized to use them, and most of the functionality is bridged between the native types and the Foundation types.
While String and NSString are mostly interchangeable, i.e, you can pass String variables into methods that take NSString parameters and vice versa, some methods seem to not be automatically bridged as of this moment. See this answer for a discussion on how to get the a String's length and this answer for a discussion on using containsString() to check for substrings. (Disclaimer: I'm the author for both of these answers)
I haven't fully explored other data types, but I assume some version of what was stated above will also hold true for Array/NSArray, Dictionary/NSDictionary, and the various number types in Swift and NSNumber
Whenever you need to use one of the Foundation types, you can either use them to type variables/constants explicitly, as in var str: NSString = "An NSString" or use bridgeToObjectiveC() on an existing variable/constant of a Swift type, as in str.bridgeToObjectiveC().length for example. You can also cast a String to an NSString by using str as NSString.
However, the necessity for these techniques to explicitly use the Foundation types, or at least some of them, may be obsolete in the future, since from what is stated in the language reference, the String/NSString bridge, for example, should be completely seamless.
For a thorough discussion on the subject, refer to Using Swift with Cocoa and Objective-C: Working with Cocoa Data Types