Most of the time I will use an exception to check for a condition in my code, I wonder when it is an appropriate time to use an assertion?
For instance,
According to this doc http://docs.oracle.com/javase/6/docs/technotes/guides/language/assert.html#design-faq-general, "The assert statement is appropriate for nonpublic precondition, postcondition and class invariant checking. Public precondition checking should still be performed by checks inside methods that result in particular, documented exceptions, such as IllegalArgumentException and IllegalStateException."
If you want to know more about precondition, postcondition and class invariant, check this doc: http://docs.oracle.com/javase/6/docs/technotes/guides/language/assert.html#usage-conditions. It also contains with examples of assertions usage.