In Java critical sections, what should I synchronize on?

As earlier answerers have noted, it is best practice to synchronize on an object of limited scope (in other words, pick the most restrictive scope you can get away with, and use that.) In particular, synchronizing on this is a bad idea, unless you intend to allow the users of your class to gain the lock.

A particularly ugly case arises, though, if you choose to synchronize on a java.lang.String. Strings can be (and in practice almost always are) interned. That means that each string of equal content - in the ENTIRE JVM - turns out to be the same string behind the scenes. That means that if you synchronize on any String, another (completely disparate) code section that also locks on a String with the same content, will actually lock your code as well.

I was once troubleshooting a deadlock in a production system and (very painfully) tracked the deadlock to two completely disparate open source packages that each synchronized on an instance of String whose contents were both "LOCK".

First, note that the following code snippets are identical.

public void foo() {
    synchronized (this) {
        // do something thread-safe
    }
}

and:

public synchronized void foo() {
    // do something thread-safe
}

do exactly the same thing. No preference for either one of them except for code readability and style.

When you do synchronize methods or blocks of code, it's important to know why you are doing such a thing, and what object exactly you are locking, and for what purpose.

Also note that there are situations in which you will want to client-side synchronize blocks of code in which the monitor you are asking for (i.e. the synchronized object) is not necessarily this, like in this example :

Vector v = getSomeGlobalVector();
synchronized (v) {
    // some thread-safe operation on the vector
}

I suggest you get more knowledge about concurrent programming, it will serve you a great deal once you know exactly what's happening behind the scenes. You should check out Concurrent Programming in Java, a great book on the subject. If you want a quick dive-in to the subject, check out Java Concurrency @ Sun