Have you asked yourself why you've always been taught that it's a good idea to make members private?
It's because (among other reasons) fields are an implementation detail. The detail of "storing data in memory", and it is an unimportant detail to any object which wishes to retrieve or set the data. Another class doesn't need to care whether he can access some memory slot somewhere - he just wants an interface for which he can pass or retrieve a value - there are the getters and setters, or properties.
Having decoupled the property from the detail of "memory based storage", we're given a large number of advantages. Primarily - we can override the behaviour of getting and setting without upsetting any code which uses the property. We can also use the property as an abstraction for retrieving data over a number of different implementations. That becomes extremely useful for testing/mocking behaviour, and providing alternative storage. If other classes depend on the implementation detail of "memory storage", you are not going to be able to change the behaviour of your class without breaking all those.
Before auto properties came along, we would typically store a field and create a getter and setter to encapsulate it for the reasons described above. An auto property automates that for us. We might write code that commonly uses fields everywhere in code, but we do so holding the idea of "I'll do this as a field for now, but that may be subject to change later if the criteria change".
Since a class knows about it's own implementation, it's usually a pointless endeavour to create private auto properties, you're not hiding the detail that's already known. Protected auto properties can be useful if you need to expose to subclasses.
As for situations where they should be avoided: When you want readonly data. (data which will not change after the object is constructed). Auto-properties lack the syntax to allow you to create an automated property that's backed by readonly data.