Imagine a computer consisting of large amounts of pebble stones, one side black, the other white, which arrangements and pattern represent what (data) the computer calculates with and also how, in which order what it is doing (program), by following a simple set of rules you can make this arrangements of stones calculate everything that's computable according to Turing – relevant XKCD: http://xkcd.com/505/ Stones like these could be called "Bits"
Now imagine that instead of being black on one side and white on the other side, those pebble stones have a special color that's flickering and changing between so fast they eye can't tell, but everytime you shine some light on it the stone stops to change its color and shows you either a solid black or white. This is called collapse of the wavefunction (short collapse) in the Kopenhagen interpretation of quantum mechanics (there's also another interpretation called "Many Worlds" but this is even more mind bending, though it makes actually desinging a quantum computer a lot easier, I'm not going to explain it here). Such stones you could call "Quantum Bits" or just Qbits.
Now you build a computer from those Qbits. You again follow the simple set of rules where each pattern of stones influences how to set the next pattern of stones. But you may not shine light on it. And because you don't shine light on them you don't know what color each Qbit stone shows. So to make the rules work you have to assume that every Qbit stone arrange may show any possible pattern, which means that with ordinary Bit stones you'd have to lay out every possible new pattern following from the previous one. But those Qbit stones are special, they're able to undergo all those varations at the same time, because of nature to always and constantly change their color until you shine some light on them: So all you have to do is lay them out in a generic pattern, which arrangement you know from the program (calculation) you want to perform and the stones will couple to the previous pattern. By doing several of those arrangements in a row each pattern of Qbits locks to the chain of patterns, slightly shifting the probability for the outcome which color they show when shining light on them.
In the last step you do something special: Not only is it possible to shine light on them that makes the stones show some color, it's also possible to shine some special light on them that forces them into a specific color. You use that special light to shine your input data (a question of sort) onto the first row of stones. And due to the cascade of Qbit patterns undergoing all the patterns possible to them every possible calculation is performed at the same time. But the overall arrangement, each step allows only for a smaller number of outcomes (it might still be a large number); with each programming step the "space" of possible answers gets smaller. Then, when you shine that special light on the first row of Qbit stones, you force all the connected Qbit stones into doing exact the one calculation that's possible for the program and the data. When you now shine that light on the last of Qbit stones, it will show you the answer to your question. And because Qbit stones can undergo all possible calculations at once they're giving you the answer very fast.
You could do the same with a computer using the good old, normal Bit stones, but for each step inbetween you had to try each and every combination and check if it matches the problem. This takes a lot of time. That's what makes Qbits special: They sort of try out everything at the same time and will arrange themself in an instant to show the right color when looked at them.
In reality the Qbits are not made from pebble stones, but from quantum systems. Everything that allows a so called Superposition that can be entangled (=connected) to other superpositions could be used: - Atoms in magnetic spin states - Electrons in magnetic spin states - Bose-Einstein Condensates either with spin-states or energy level superpositions - Magnons in solidd and so on. It's a really long list because about everything in nature has some so called quantum numbers attached to it, that could be used. This special light used to program the computer are very stable lasers.
So why don't we just build quantum computers. Well, there's a catch: The light you shine on to look at it, everything not specially prepared acts as this kind of light. Some freak radio wave coming by: Collapses the Qbits. Some impurity, like some stray atom in the vacuum chamber: Collapse. A glitch in your programming lights (lasers): Collapse. And the collapse always engulfs the whole quantum state, not just a single Qbit.
And if the Collapse happens, before you were able to lay out your program and data it messes up your computer and you have to restart the programming from scratch. And the longer and complex your quantum program and data is, the more Qbits you have to use, the harder it gets to prevent the undesired collapse to happen.
To prevent the collapse you have to get your quantum system really, really, really clean, you must cool it down as far as possible, you must shield it from any undesired thing coming from outside. You need very, very, very good lasers and power supplies. You need a table that's so solid that it lets no vibration through. In short: It's really complicated to build quantum computers right now.
The hope is, that we'll find ways to make Qbits, that don't collapse that easily, that ideally only interact with what we've designed them to interact with.
Quantum computing is another way of processing data. Much like your computer has a sound card and a graphics card, computers will one day have quantum cards for solving problems known as "intractable." These intractable problems are very difficult for normal computers to solve because they require looking at all the possible configurations of the solution, so as the problem size grows, the amount of time required to test all the configurations becomes unmanageable. Quantum computers are able to set up a state in which all possible configurations exist at the same time by overlapping quantum states which are both infinite and discrete. Once this situation is created, the Quantum state anneals or settles to only the configurations that are solutions to the problem. This currently can only be done for certain types of problems and the gains in speed are only seen when the problems are extremely large.