Speedcubing is the art of solving a Rubik's cube as quickly as possible. The procedure is as follows:
 The cube must be scrambled using a 25 move random sequence. As any cube can be solved with just 21 moves, this should be enough to ensure that the solve is not too easy.
 The cube may be inspected for 15 seconds prior to the solve.
 Solve the cube.
 When the cube is put back down, it must be solved or be no more than an eighth of a twist away.
Because the chances of having a lucky solve are fairly high, a single solve does not mean much. So competitions usually involve a minimum of five cubes of which an average is taken. The speedcubing community being fairly small, the worldwide accepted rating is the average of the middle ten of twelve cubes.
How quickly is quickly?
I can solve a cube in about 25s when taking an average of 10 cubes. The best in the world can do this in under 15s. The official world record for one cube stands at 10.48s and is held by Toby Mao from the USA (August 2011: this is no longer the case, see link below). The best nonlucky solve currently stands at around 10s and lucky solves can be even quicker. Because of this, it is just a matter of time before someone breaks the world record again.
Where the limits of the average lie is another matter. With the current techniques, it is possible to solve the cube in about 50 moves ; do this at 3 moves per second and you have a cube solved in 17s. The fastest cubers seem to be on 34 moves/s for the first 30 moves and then 67 moves/s for the last 20, allowing times of 1011s. As we will see, it is unrealistic to learn a solution which averages less than 40 moves, so assuming that the speed of execution will not increase by much (an assumption I already made once which turned out completely incorrect), 89s seems to be the inferior limit to which the average times will go. (This estimate revised on 24th oct 2007  previous estimate was 12 seconds based on not being able to better 34 moves per second as an overall average. Another review august 2011: it seems the speed limits are way faster than I thought in terms of moves per second).
I can't even solve a Rubik's cube  how will I get that fast ?
A beginner's method to solving the cube comprises about 100 moves. The solution will be made up of several phases and for each phase, you will need to both learn some algorithms and when to apply them. Although it is easy to perform 100 moves at a rate of 1 move/s, you will , at first, be spending as much time deciding which algorithm to apply as you will actually applying it.
Don't be scared. Once you start understanding the mechanics of the cube, a beginner's solution only has about 5 algorithms which you actually need to learn. The rest is non trivial, but once you know why it works, you no longer need to memorise it.
Once you have a solution memorised and you are no longer fiddling through the booklet and trying to transform weird notation into physical moves, three minutes is quite an easy target. Unfortunately, the time taken to choose what you do next will almost never take less than 2 seconds. Most beginner's solutions have about twenty places where you have to pause and make a decision. So even going at world class speed, it will take over a minute to solve a cube.
This means that the most important thing is to learn a solution to which you can add short cuts so as to progressively move down to 60 moves. Next, you must learn to look ahead ; most algorithms only affect five or six pieces so while you are performing one algorithm (this becomes easy to do without thinking), you can already be preparing what you will need to do next. All solutions break down into two parts: the first is quite intuitive and, as you will only be working on solving one or two pieces at a time, this can be done without almost no pausing ; the second usually involves between one and four longer algorithms during which there will be little point in watching the pieces as the choice of the next algorithm will be dictated by up to eight of them.
Although being able to perform algorithms quickly is not the most important skill and in some cases actually hinders your ability to look ahead, here are a few pointers:
 You need a good cube which is well lubricated. If you can't turn faces round with your pinkie, you need more lube. Try silicon spray.
 A trigger is a move which makes two or more turns in one smooth flick of the wrist. To get the idea, try holding the cube in your left hand with fingers well clear of the top and right slices. Now place the right index, middle and ring fingers on the back side of the right slice and your thumb on the middle facelet of the front of the right slice. If you pull your index finger towards you, this should turn the top slice clockwise ; Just before the slice becomes aligned again, twist your whole wrist so as to turn the right slice counterclockwise. The flick of the wrist snaps the top slice into place and simultaneously turns the right slice.
 Break up long algorithms into sequences of two and three move triggers. This gives the added bonus of easing memorisation of the algorithms.
What is a good solution method for speed?
There are three basic solution flavours: those that start by solving all the corners, appropriately called Corners First; those which solve the cube layer by layer of which the best known and most widely used is the Fridrich method; and the Petrus method, which starts by building a 2x2x2 corner and then expands around that. A number of other methods mix these together in various aspects. I have little knowledge of the Corners First method so, although it allows worldclass times, I will only discuss the advantages and disadvantages of the other methods.
In a beginner's layer by layer solution, you first form a cross with the edges of the top face, then solve the corners of the top face, then the edges of the middle face, then solve the last layer. The Fridrich method starts with the cross, but on the bottom face, then solves the four corneredge pairs, a pair being a corner and the edge piece of the middle layer which goes above it. It then solves the last layer by first orienting all the last layer pieces and then permuting them to finish the cube.
The Petrus method starts by solving one corner and the three edges adjacent to it. This forms a small cube comprising the corner, the three edges and three centerpieces. This is then extended to a 2x2x3 block by adding on another corner and its corresponding edges. All the edges are then oriented and the first two layers are completed without disturbing this orientation. The last layer is solved by placing the corners, orienting the corners and finally placing the edges (they are already orientated).
Both the Fridrich and Petrus methods solve the first two layers and the last layer separately. This has resulted in every hybrid having been tried out by someone. For the first two layers, any combination will be too complex to explain here, just pick what feels right for you: Fridrich takes more moves but there is less variation; Petrus takes fewer moves but it is more difficult to extract frequent patterns. For the last layer, there are four things to be achieved: You have to orient and place both corners and edges. This can be done in between two and four steps. Solving the whole last layer in one step would require memorising over a thousand algorithms so this is clearly not feasible. The advantage of Petrus is that a 30 move lastlayer is possible with only four algorithms. The advantage of the Fridrich method is that it is easy to recognise orientations (just look at the last face and see whether the colors match with the center) and permutations are easiest to identify once all the pieces are oriented; you will, however need at least ten algorithms for a basic last layer. A two step last layer requires knowing over eighty algorithms. This is a lot. This is also not necessary, even to get times below 20s.
That's all very well, but what method should I use?
In the long run, you will probably experiment with various methods and pick the choices you like best. Here is what I think is the best way of learning to solve a cube with the aim of solving for speed, bearing in mind that a small number of algorithms is important at first, while a reduced number of steps and ease of recognition become important later on. The first method starts out at an average of eighty moves and each subsequent method reduces that number to eventually arrive at a full Fridrich solution of fifty moves.
 Solve the cross on the bottom, then solve three of the bottom corners. By rotating the bottom slice, you can use the empty corner to insert three of the edges of the middle layer. Orient the last layer edges, position the last layer corners, orient the last layer corners and finally position the last layer edges.
 Swap the positioning and orienting of last layer corners. You will then know half of the algorithms required to simultaneously position corners and edges.
 Position the last layer in one algorithm instead of two
 At this stage, you know only one algorithm for orienting the last layer corners. Learn the remaining six, bearing in mind that you no longer need to preserve corner permutation.
 By this time, you should feel comfortable enough putting the last corneredge pair in together to be able to do the same with each of the other pairs.
 If you really want to, you can learn the algorithms for orienting all the last layer pieces in one step.
I have no intention of integrating that last step. There are way too many algorithms to learn in my opinion. Instead, I orient the edges before inserting the last corneredge pair and then insert it while preserving edge orientation.
What other tips can you give me?
 Any beginner's method has the solved part nearest to you ; this is comforting, but does not allow you to see the unsolved pieces easily. Keep the solved part away from you by solving the cross on the bottom and having the last layer on the top.
 Use the inspection time to solve the whole cross in your head if possible. Then solve the cross slowly while looking for the pieces needed for the next step.
 Don't solve a corner edge pair too quickly. Use the time to look for the next pair.
 Try practising to a metronome
 Try recording your solve with a microphone to be aware of pauses. It is very easy to learn to perform algorithms quickly, but difficult to solve without long pauses.

Know the colors of your cube. Always solve the cross on the same face.

It's just a cube. Have fun.
Visit
www.speedcubing.com for all things cuberelated.
The Petrus method can be found at
www.lar5.com