Backhand...
is the most common and easiest way to throw a frisbee. Grip the disc with your thumb on the top, your index finger along the outside edge (rim), and the disc resting on the rest of the fingers that are held spread out on the bottom of the disc. This is the most common grip, known as the "pinch grip" for simple, not so powerful and long, throws.

In long disc golf "drives", the so called "power grip" is better suited. See the flying disc academy section on disc golf for more advanced lessons on the backhand throw.

Assuming you are right handed, position yourself with your right side towards where you want to throw. Keep your feet at about shoulder width from eachother.

One common beginners mistake is that you want to turn your body to get your chest directly facing the target (or your catching partner).
 

Grip the disc like this...

Now, draw your arm back a bit, bend your wrist slightly, and then pull the disc forward in a straight line, in paralell with your chest in the intended direction of the throw. Let the disc pass your chest completely beofre releasing it.

Keep your right side pointing towards the target...

The wrist should help to impart spin to the disc just before the release. When you pull the disc in a straight line, your wrist should automatically bend in front of your body, and then extend in the release to produce the disc spin.
 

Don't stop the throwing motion immediately after releasing the disc. Continue in a follow through. This will make your "swing" smoother and the throw easier to control. It also helps preventing injury.

By tilting the outside edge of the disc slightly down (for a left turning throw) or slightly up (for a right turn) you will with some practise be able the make controlled curve throws.

If the disc flutters and turns heavily right, you have probably failed to give the disc enough stabilizing spin in relation to its forward speed.

Try not to keep your arm and wrist all tense through the throw, bend the wrist a little more, and tilt the outside edge of the disc slightly down. As the engraving on the early frisbees said: "Flat flip flies straight, tilted flip curves. Experiment"
 

 Flying Disc Academy:
Disc Golf: The Drive
 

Forehand...
For a forehand (also known as sidearm) throw, asssumed you're right handed, grip the disc on the right side of your body, with your thumb on the top, your middle finger on the inside rim, and your index finger as support on the bottom side of the disc.

Your ring finger and pinkie are actually not used in this throw, and should remain curled up against your palm.

Position your body at a 45 degree angle towards the target...

Grip like this. Note that it's the pad of your middle finger that should rest against the inside rim, not the side of the finger...

Stand with your left foot a bit in front of your right one, at about shoulder width. (So your chest is at about 45 degrees angle towards the direction you want your throw to fly. Your weight should initially be mostly on your back (right) leg.

When throwing regular frisbees, note that your forearm should not be parallel to the ground. The forearm, and the disc, should point downwards to the ground at an angle around 30-40 degrees.

(When throwing heavy golf discs for distance, your arm should be higher though. More on that in the disc golf classes in the flying disc academy.)

Now pull your throwing arm backwards, until the disc is further away from the target than your right leg, and lower your right shoulder a little bit.

The throw is initiated by a weight shift from your rear (right) leg to the front (left) leg. Push your right elbow forward and let the disc go with a smooth flick of the wrist.

In a forehand throw it's mainly the wrist that gives the disc the spin and forward momentum, so do not swing like you would in a forehand tennis hit. Just a simple snappy flick does the trick. Release the disc when it's just in front of your body.

At the wrist flick, the inner part of your wrist should turn upwards (i.e. clockwise) in the same manner as if you were going to splash water from the pool in someone's face. It's very important that the outer edge of the disc is still pointing downwards at the wrist flick.

 discgolferusa.com:
Throwing Sidearm