Properties and ivars are two completely different things.
And instance variable is a variable stored inside the object, so each instance has its own. It is referenced by pointer addition relative to the object pointer/self (slightly indirected for the modern runtime, but functionally equivalent). ivars are generally internal to a class, and by default can only be accessed by the class and its descendents (@protected). Within methods they are available with no qualification, otherwise they can (but rarely are, ad usuaually should not) be accessed via indirection, eg obj->ivar.
A property defines a getter and setter (the setter is optional) interface. That's all it does. It defines two public methods:
- (TYPE) propname;
- (void) setPropname: (TYPE) newPropname;
These are defined as methods exactly as if you declared them like that, no more, no less. These methods are called either with the normal syntax ([obj propname] and [obj setPropname:n] or using the modern dot notation (obj.propname or obj.propname = n). These two options are syntactically different only, they behave identically, and you can use dot notation whether the methods are declared with @property or declared manually as above.
You must then implement the methods in the implementation, either by writing the methods yourself, by using @synthesize, or by handling the missing method dynamically.
Properties may be backed by an ivar (named the same or named differently (my preference to avoid confusion)), or they may not. They may store their value elsewhere, or they may calculate it from other data.
For example, you might have:
@property (nonatomic, readonly) NSString* fullname;
and then implement - (NSString*) fullname to return the concatenation of firstname and lastname.