Adapted from a forum posting originally contributed by rhino of GoodOldGold.com, and others
Here is a simply little test anyone can perform to help determine if what they are looking at is a real diamond or a CZ.
First, take a blank piece of paper and make a dot on it with a pen:

Next, take the stone in question and turn it upside down on the paper. We will be positioning the diamond, table down and centered on the dot.

When you center the gemstone on the dot...

In a CZ you will see a circular reflection of that dot clearly through the pavilion. The stone on the left is the CZ. The stone on the right is a real diamond. A diamond will break up that reflection to the point where it is almost imperceptible. In a CZ it will be very easy to see.
There are other ways of making the distinctions between fakes and real but this is an easy simple test that anyone can perform.
Unless you have special equipment, determining whether a mounted diamond is real or not is much harder than a loose one. However, there are a few things you can do to help 'ease your mind' - though they are NOT as accurate as the paper test.
The photo below shows a stone that's been around a bit. In this photo you can easily make out scuffs/abrasions at the facet junctions as well as scratches on the table.

Here is a CZ alongside an AGS ideal cut to demonstrate the differences. Also, notice that CZ's are generally cut pretty sloppy when it comes to symmetry characteristics.

Weight. If you have a precision scale, another great test for an unmounted CZ/diamond is weight. A CZ of the same size as a diamond will weigh nearly twice as the diamond. Remember, a 1.0ct (real) diamond weighs 200 milligrams.
Gizmos. The easiest way to reliably test is to use a diamond tester. The typical tester measures thermal and electrical conductivity. The more advance testers also measure reflectivity, and the top-of-the-line testers check both.
The following so-called "tests" are not accurate ways to tell fake from real diamonds:
Take a trip to your friendly neighborhood jeweler (preferably a real jeweler and not a chain at the mall) with the gemstone in question. Most jewelers walk around with a diamond tester and a loupe in their pocket, and will gladly check your gemstone without charging you.