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;  }}

Declaring a variable as volatile means that modifying its value immediately affects the actual memory storage for the variable. The compiler cannot optimize away any references made to the variable. This guarantees that when one thread modifies the variable, all other threads see the new value immediately. (This is not guaranteed for non-volatile variables.)

Declaring an atomic variable guarantees that operations made on the variable occur in an atomic fashion, i.e., that all of the substeps of the operation are completed within the thread they are executed and are not interrupted by other threads. For example, an increment-and-test operation requires the variable to be incremented and then compared to another value; an atomic operation guarantees that both of these steps will be completed as if they were a single indivisible/uninterruptible operation.

Synchronizing all accesses to a variable allows only a single thread at a time to access the variable, and forces all other threads to wait for that accessing thread to release its access to the variable.

Synchronized access is similar to atomic access, but the atomic operations are generally implemented at a lower level of programming. Also, it is entirely possible to synchronize only some accesses to a variable and allow other accesses to be unsynchronized (e.g., synchronize all writes to a variable but none of the reads from it).

Atomicity, synchronization, and volatility are independent attributes, but are typically used in combination to enforce proper thread cooperation for accessing variables.

Addendum (April 2016)

Synchronized access to a variable is usually implemented using a monitor or semaphore. These are low-level mutex (mutual exclusion) mechanisms that allow a thread to acquire control of a variable or block of code exclusively, forcing all other threads to wait if they also attempt to acquire the same mutex. Once the owning thread releases the mutex, another thread can acquire the mutex in turn.

Addendum (July 2016)

Synchronization occurs on an object. This means that calling a synchronized method of a class will lock the this object of the call. Static synchronized methods will lock the Class object itself.

Likewise, entering a synchronized block requires locking the this object of the method.

This means that a synchronized method (or block) can be executing in multiple threads at the same time if they are locking on different objects, but only one thread can execute a synchronized method (or block) at a time for any given single object.

volatile:

volatile is a keyword. volatile forces all threads to get latest value of the variable from main memory instead of cache. No locking is required to access volatile variables. All threads can access volatile variable value at same time.

Using volatile variables reduces the risk of memory consistency errors, because any write to a volatile variable establishes a happens-before relationship with subsequent reads of that same variable.

This means that changes to a volatile variable are always visible to other threads. What's more, it also means that when a thread reads a volatile variable, it sees not just the latest change to the volatile, but also the side effects of the code that led up the change.

When to use: One thread modifies the data and other threads have to read latest value of data. Other threads will take some action but they won't update data.

AtomicXXX:

AtomicXXX classes support lock-free thread-safe programming on single variables. These AtomicXXX classes (like AtomicInteger) resolves memory inconsistency errors / side effects of modification of volatile variables, which have been accessed in multiple threads.

When to use: Multiple threads can read and modify data.

synchronized:

synchronized is keyword used to guard a method or code block. By making method as synchronized has two effects:

1. First, it is not possible for two invocations of synchronized methods on the same object to interleave. When one thread is executing a synchronized method for an object, all other threads that invoke synchronized methods for the same object block (suspend execution) until the first thread is done with the object.

2. Second, when a synchronized method exits, it automatically establishes a happens-before relationship with any subsequent invocation of a synchronized method for the same object. This guarantees that changes to the state of the object are visible to all threads.

When to use: Multiple threads can read and modify data. Your business logic not only update the data but also executes atomic operations

AtomicXXX is equivalent of volatile + synchronized even though the implementation is different. AmtomicXXX extends volatile variables + compareAndSet methods but does not use synchronization.

Related SE questions:

Difference between volatile and synchronized in Java

Volatile boolean vs AtomicBoolean

Good articles to read: ( Above content is taken from these documentation pages)

https://docs.oracle.com/javase/tutorial/essential/concurrency/sync.html

https://docs.oracle.com/javase/tutorial/essential/concurrency/atomic.html

https://docs.oracle.com/javase/8/docs/api/java/util/concurrent/atomic/package-summary.html