Flying the Disc

Flying discs (including Frisbees) can be thrown in many ways. All involve spinning the disc to give it gyroscopic stability, and accelerating its mass to a certain velocity. Without spin, a disc will wobble and fall; without velocity, the disc will not go anywhere. Using these two guidelines, any number of throws are possible. Most discs are designed to create lift when thrown with the flat side up.
Techniques.
Right-side up.
Right-side up throws are all similar in that they react the same way to the tilt of the disc when it is released. A disc thrown right-side up will accelerate in the direction of the low end of the disc. A disc tilted leading-edge up will lose speed at the end of the throw and make a gentle landing; if tilted sideways (known in aeronautics terms as roll), it can curve around objects.
There is a language for describing throws that curve. Both descriptions are relative to the direction the person is facing and intends to throw. This axis is marked in "red" in the picture.
Backhand.
This is probably the most commonly learned throw, and also one of the most powerful. There is a long windup for the throw, both in terms of path length and time.
Variations.
The backhand is an excellent general-purpose throw. Several modifications have been created; some are listed here.
Forehand.
This throw (also known as the "flick" or the "two-finger", or the "side-arm" in the UK) is a staple of the disc fan's repertoire, as well as the bread-and-butter throw of Ultimate players. Focused in the wrist, this throw takes little time to execute.
Note that 90% of this throw is below the elbow, and most of that at the wrist. A common mistake is to attempt to use the upper arm and shoulder to add power to the throw. Usually, this results in little or no spin being imparted on the disc, which causes it to fall quickly. It's also common for the flick of the wrist to pull the outside edge of the disc up; to compensate, hold the disc somewhat loosely, so that the outer edge hangs down slightly. Experiment with the cocking and flicking motions to determine how best to produce a level throw.
As you learn how to throw a flick, it may help to tuck your elbow in towards your body; this is a bad habit to get into if you play Ultimate, but may assist with preventing you from using too much shoulder and arm. As you improve, work on extending your arm out away from your body. Advanced throwers will note that the arm and wrist action is much like a bullwhip cracking sideways; the addition of whip-like motion in your upper arm, shoulder, and even upper body (by rotating the hips) can impart more power onto your throw, but is difficult to control.
Variations.
The forehand is an extraordinarily versatile throw, and can be adapted to many different situations.
Push Pass.
The Push Pass: A little-used variant of the forehand, it is thrown with a grip similar to a backhand (index finger on the outer rim of the disc, thumb on top, fingers curled underneath) but is released on the forehand side. The wrist "pushes" the disc forward while spin is imparted "backwards" by rolling the disc off the index finger. A final flick of the index finger finishes the release. Frequently, very little spin is actually imparted, which makes this throw tend to flutter and fall more often than not. However, with practice the thrower can add much more spin to this throw and it may become useful over short distances. It is also useful in Ultimate as the typical form of this throw can be placed in the region between a defender's body and extended arms.
Beach Thumber.
The Beach Thumber (also known as a Peach) is unpopular in Ultimate circles due to its unpredictable flight path and proximity of release to the thrower's body. Its primary advantage is that it can be a very hard, very accurate throw, and therefore useful for those uncomfortable with the staple throws of the game, the forehand and backhand. 
Overhand.
The Overhand (also known as the "waffle", "discus", "wrist-hook", "chicken wing", or "biscuit" in the UK) varies in popularity among ultimate players, because the alternative, the traditional forehand, allows greater sideways arm extension, useful in moving the disc around defenders. The Overhand is most useful when the disc is caught above the head and must be thrown quickly without changing grip. It is similar to the "Hammer" in that release typically (but not always) occurs above shoulder level, but comes with a different set of drawbacks. This throw is often used in attempts at The Greatest.
The Overhand is a situationally dependent throw. It can be extremely useful in the right situation, especially when there is no time to change grip. But the fact that the thrower typically must lean in towards the defender on release hampers its effectiveness for ultimate. This throw is popular for advanced catch and freestyle, but some ultimate players deride it as the "Useless", the "Push Noodle", or even the "Nazi Throw" as when the throw is finished, the arm is in the Nazi salute.
Duck.
The Duck (also known as a "bear claw", a "duder," or a "useless") is thrown with a similar grip to the waffle, except it is the backward version of it. While the waffle is thrown with counter-clockwise spin (for right handers), the duck is thrown with clockwise spin. It is usually thrown with the arm out to the side or above the head. It is called the duck due to the shape of the gripping hand during the throw, as if making a duck shadow puppet. This throw is used in attempts at The Greatest (jumping out of bounds and throwing the frisbee back in to play while in the air).
Upside-down.
A disc thrown upside-down has a very different flight path than one thrown right-side up. Lift is applied towards the rounded side of the disc; if the disc is completely upside-down, the lift is now propelling the disc straight into the ground. Thus, these throws typically have a stronger arc to them, tend to bank or roll as they fly, and are much harder to catch. Where a right-side up throw tends to spend a considerable portion of its flight path at a catchable height, an upside-down throw tends to land abruptly at a certain point.
To facilitate discussion, the following convention is used: a disc that is level and flying right-side up is considered to be flying at 0 degrees relative to the ground (DRtG). A disc that is level and flying upside-down is considered to be flying at 180 DRtG. A disc that is flying exactly vertically would be at 90 DRtG, etc.
Hammer.
The Hammer is one of the more difficult throws to master, despite being gripped just like a normal forehand throw, and the throwing motion being relatively natural. Hammers are difficult to aim well due to their high, arcing flight path, and their tendency to catch any amount of wind.
A hammer, when thrown properly (and in no wind) by a right-handed thrower, will arc up and to the left as it moves away from the thrower. As it flies, the disc will roll over until it is completely upside-down (180 DRtG), falling down and to the right again. Ideally, it should reach its target at between 135 and 180 DRtG to make catching it as easy as possible.
However, hammers are notoriously difficult to throw "properly". Common failure modes include:
A Blade can be corrected by increasing the angle of release (closer to 180 DRtG); a Double Helix can be corrected by decreasing it. Experimenting with the angle of release should produce a happy medium which works for the individual thrower. Angle of release will also affect the horizontal travel of a hammer, which is difficult to judge even in the lightest breeze. The near-vertical release of the hammer makes them highly susceptible to crosswinds, blowing them well off target. Hammers rely on spin for stability even more than most throws; a hammer thrown with insufficient spin will flop about as it flies, making the receiver's job even more difficult.
A hammer, like most throws, can be released with any amount of angle and pitch and at any strength. A hammer released at nearly-180 DRtG and straight ahead (i.e. with a pitch close to 0) will stay mostly flat and fly directly at a receiver. This throw is difficult to block (in Ultimate) but tends to arrive very fast and at an unpredictable angle, since it often flies like a Double Helix. Hammers are often last ditch efforts in a game of Ultimate when the thrower has reached a stall count of 8 or 9 and needs to get rid of the disc; either as a "punt" or "hail mary" to a deep in the end zone. (See also the Scoober, below.) A Blade can also be thrown intentionally; this is commonly called a Roller in disc golf, as they tend to hit the ground and roll for long distances if thrown very near 90 DRtG.
Wheel of Death.
The Wheel of Death (also known as The Corkscrew, The Bammer and The Bowler) is the “backhand version” of the hammer. It is a very difficult throw to master, and not very useful. The throwing motion is nearly as unnatural as possible. The wheel of death inherits all of the limitations and challenges of the hammer, and it has a few of its own.
The wheel of death, when thrown properly (and in no wind), should fly in a path entirely similar to that of the hammer, only with left and right directions interchanged; this is, perhaps, its only advantage. The wheel of death has many more disadvantages than advantages, however. These disadvantages include:
Furthermore, the wheel of death can fail in all of the ways a hammer does, plus, if any upward pressure is placed on the trailing edge of the disc, the throw will air-bounce downwards, resulting in a highly deceptive flight path and a very short pass. Due to its myriad of deficiencies, the wheel of death is not generally implemented as anything more than a curiosity.
Scoober.
Another upside-down variant of the forehand, the scoober (also known as the "Spoon pass" or "Hiawatha") is similar to a hammer, but thrown over the left shoulder for a right-handed thrower instead of over the head. The scoober is an advanced throw that travels in a path similar to the hammer, although it cannot be thrown as far because it relies almost exclusively on the wrist for power. The scoober can be an effective short-range (10 to 20 yards/meters) throw and is used in ultimate for breaking the mark and against zone defenses.
A variant of this throw is termed the blind scoober. Rather than being thrown over the left shoulder (by a right-handed thrower), a blind scoober is thrown backwards with an outstretched arm without ever looking to see where the intended receiver is. Though rarely successful, the comedic value of the throw is unquestionable.
Other throws are also known as a scoober; these seem to vary by geography. The hammer is sometimes referred to as a scoober.
Thumber.
The Thumber (not to be confused with the "Beach Thumber") is an advanced throw that is rarely used in competitive play, compared to the Hammer or standard forehand. It has a flight path that is the mirror-image of the Hammer (arcing high and to the right for a right-handed thrower). It can be useful when the disc needs to drop quickly and fly with an opposite helix to the Hammer.
Viggiano/Underhand Hammer.
The "Viggiano" is a throw without much power, but it is remarkably stable, considering its aerobatic nature (because the disc ends up in a 0 DrtG orientation). Its initial flight path is similar to that of the hammer, but opposite (flipped 180 degrees) about a plane that is horizontal relative to the ground. When thrown correctly, the disc will initially fly at about 80 degrees relative to the ground, but will turn right side up in mid-flight and remain as such, flying much like a weakly-thrown backhand after the turn. The short-windup throw is useful for short-distance passes only, but is especially helpful if no one else, including a blocker, expects it. For instance, if another player is trying to block your throw, you may be able to throw the disc underneath and around his or her arms. Typically, in competitive ultimate, however, this throw seeing that it is released to close to the body is often frowned upon unelss the thrower is being marked very tightly. It is very easy on the joints to throw.
Variation: Reverse Viggiano: This throw is similar to the Viggiano, except one starts with the top of the disc pointing away from the side of the body, with the thumb underneath and two fingers on top. Like the Viggiano, the disc will initially have a vertical orientation (about halfway between right side up and upside-down), with the top of the disc very slightly facing the sky (280 DrtG) and turn in mid-flight, flying right-side up (0 DrtG) the rest of the way. The throw is different in that one twists his or her wrist in the opposite way prior to release (i.e., counter-clockwise, as though one is loosening a typical-thread screw) and flicks his or her wrist toward the front of the body to impart spin, much in the same way a cellist flicks his or her wrist with the bow hand toward the cello while playing.