'copy' will cause the setter for that property to create a copy of the object, and is otherwise identical to strong. You would use this to make sure that if someone sets your property to a mutable string, then mutates the string, you still have the original value. If the string isn't mutable, Cocoa will silently optimize out the copy operation, which is nice :)
'strong' will keep the property's value alive until it's set to something else. If you want incoming mutable strings to change out from under you (not impossible, but not all that common, a thing to want), then strong would be the right thing to do. Generally strong is more useful for objects that represent something more complex than a simple "value" (i.e. not NSString, NSNumber, NSValue, etc...).
'assign' is the default (and indeed only) possible setting for an integer. Integers can't be retained or copied like objects.