The last two are identical; "atomic" is the default behavior (note that it is not actually a keyword; it is specified only by the absence of nonatomic -- atomic was added as a keyword in recent versions of llvm/clang).

Assuming that you are @synthesizing the method implementations, atomic vs. non-atomic changes the generated code. If you are writing your own setter/getters, atomic/nonatomic/retain/assign/copy are merely advisory. (Note: @synthesize is now the default behavior in recent versions of LLVM. There is also no need to declare instance variables; they will be synthesized automatically, too, and will have an _ prepended to their name to prevent accidental direct access).

With "atomic", the synthesized setter/getter will ensure that a whole value is always returned from the getter or set by the setter, regardless of setter activity on any other thread. That is, if thread A is in the middle of the getter while thread B calls the setter, an actual viable value -- an autoreleased object, most likely -- will be returned to the caller in A.

In nonatomic, no such guarantees are made. Thus, nonatomic is considerably faster than "atomic".

What "atomic" does not do is make any guarantees about thread safety. If thread A is calling the getter simultaneously with thread B and C calling the setter with different values, thread A may get any one of the three values returned -- the one prior to any setters being called or either of the values passed into the setters in B and C. Likewise, the object may end up with the value from B or C, no way to tell.

Ensuring data integrity -- one of the primary challenges of multi-threaded programming -- is achieved by other means.

Adding to this:

atomicity of a single property also cannot guarantee thread safety when multiple dependent properties are in play.

Consider:

 @property(atomic, copy) NSString *firstName;
 @property(atomic, copy) NSString *lastName;
 @property(readonly, atomic, copy) NSString *fullName;

In this case, thread A could be renaming the object by calling setFirstName: and then calling setLastName:. In the meantime, thread B may call fullName in between thread A's two calls and will receive the new first name coupled with the old last name.

To address this, you need a transactional model. I.e. some other kind of synchronization and/or exclusion that allows one to exclude access to fullName while the dependent properties are being updated.

You are specifically asking about how they internally work, so here you are:

No synchronization

private int counter;

public int getNextUniqueIndex() {
  return counter++; 
}

It basically reads value from memory, increments it and puts back to memory. This works in single thread but nowadays, in the era of multi-core, multi-CPU, multi-level caches it won't work correctly. First of all it introduces race condition (several threads can read the value at the same time), but also visibility problems. The value might only be stored in "local" CPU memory (some cache) and not be visible for other CPUs/cores (and thus - threads). This is why many refer to local copy of a variable in a thread. It is very unsafe. Consider this popular but broken thread-stopping code:

private boolean stopped;

public void run() {
    while(!stopped) {
        //do some work
    }
}

public void pleaseStop() {
    stopped = true;
}

Add volatile to stopped variable and it works fine - if any other thread modifies stopped variable via pleaseStop() method, you are guaranteed to see that change immediately in working thread's while(!stopped) loop. BTW this is not a good way to interrupt a thread either, see: How to stop a thread that is running forever without any use and Stopping a specific java thread.

AtomicInteger

private AtomicInteger counter = new AtomicInteger();

public int getNextUniqueIndex() {
  return counter.getAndIncrement();
}

The AtomicInteger class uses CAS (compare-and-swap) low-level CPU operations (no synchronization needed!) They allow you to modify a particular variable only if the present value is equal to something else (and is returned successfully). So when you execute getAndIncrement() it actually runs in a loop (simplified real implementation):

int current;
do {
  current = get();
} while(!compareAndSet(current, current + 1));

So basically: read; try to store incremented value; if not successful (the value is no longer equal to current), read and try again. The compareAndSet() is implemented in native code (assembly).

volatile without synchronization

private volatile int counter;

public int getNextUniqueIndex() {
  return counter++; 
}

This code is not correct. It fixes the visibility issue (volatile makes sure other threads can see change made to counter) but still has a race condition. This has been explained multiple times: pre/post-incrementation is not atomic.

The only side effect of volatile is "flushing" caches so that all other parties see the freshest version of the data. This is too strict in most situations; that is why volatile is not default.

volatile without synchronization (2)

volatile int i = 0;
void incIBy5() {
  i += 5;
}

The same problem as above, but even worse because i is not private. The race condition is still present. Why is it a problem? If, say, two threads run this code simultaneously, the output might be + 5 or + 10. However, you are guaranteed to see the change.

Multiple independent synchronized

void incIBy5() {
  int temp;
  synchronized(i) { temp = i }
  synchronized(i) { i = temp + 5 }
}

Surprise, this code is incorrect as well. In fact, it is completely wrong. First of all you are synchronizing on i, which is about to be changed (moreover, i is a primitive, so I guess you are synchronizing on a temporary Integer created via autoboxing...) Completely flawed. You could also write:

synchronized(new Object()) {
  //thread-safe, SRSLy?
}

No two threads can enter the same synchronized block with the same lock. In this case (and similarly in your code) the lock object changes upon every execution, so synchronized effectively has no effect.

Even if you have used a final variable (or this) for synchronization, the code is still incorrect. Two threads can first read i to temp synchronously (having the same value locally in temp), then the first assigns a new value to i (say, from 1 to 6) and the other one does the same thing (from 1 to 6).

The synchronization must span from reading to assigning a value. Your first synchronization has no effect (reading an int is atomic) and the second as well. In my opinion, these are the correct forms:

void synchronized incIBy5() {
  i += 5 
}

void incIBy5() {
  synchronized(this) {
    i += 5 
  }
}

void incIBy5() {
  synchronized(this) {
    int temp = i;
    i = temp + 5;
  }
}