Quantum entanglement is the curious property of subatomic particles which means that two or more particles which have interacted in the past exhibit surprisingly strong correlations when various measurements are made on them. The exact nature of these correlations has been the source of a huge amount of debate in physics-philosophy circles, although quantum physicists have generally been quite happy to work with them on a practical level where necessary without worrying too much about what might actually be going on. It looks, superficially at least, as if the correlations observed must be the result of the particles exchanging information with each other faster than light, which any physicist or science fiction fan will tell you is equivalent to information being sent back in time.
However, it remains a matter for debate whether any information exchange is really taking place. You see, the nature of the connections is such that in practice it is quite impossible to send any message faster than light; we don't, and theoretically cannot, exert sufficient control over the interactions involved to do so. This makes it impossible to send a message to your past self, ruling out all the obvious causal paradoxes of that sort; however, it is not clear that paradoxes which don't rely on intentional communication can be ruled out so easily. In any case, quantum entanglement has the rather odd property that while some kind of 'communication' is often said be taking place between two widely-separated receivers, nothing that anyone does at either end will make the slightest bit of difference to what happens at the other. David Deutsch and Patrick Hayden argue persuasively in a 1999 paper1 that there is no reason to suppose that any information is really being transmitted faster than light; they propose, instead, that
...what has been mistaken for nonlocality is the ability of quantum systems to store information in a form which, like a cyphertext, is accessible only after suitable interactions with other systems.
Whatever the exact nature of the information-sharing involved in entangled systems, it has become clear that it is ubiquitous in the sub-atomic realm, and it has a number of very promising practical applications (none of which, by the way, makes any use of anything like superluminal communication). One such application is quantum cryptography - the possibility of sending an encrypted message by means of entangled particles. Knowing about the correlations between two entangled particles makes it possible, at least in theory, to use them to send a coded message. This technique should in principle be completely secure, since if anyone were to intercept the photon en route, the recipient would be able tell. Briefly, the technique works by sending a stream of photons to the recipient which are all entangled with photons measured by the sender. By comparing the measurements they made over a normal phone line, say (a string of symbols which is meaningless without the results themselves), they are able to send messages to each other which cannot possibly be intercepted without their knowledge.
Another promising application for quantum entanglement is in quantum computing, the nascent field which may one day replace digital computing entirely, but which so far has not yet led to anything of practical use. Most, but not all2, ideas of quantum computing rely on entanglement, making use of 'entangled qubits' (also known as e-bits). A qubit is the quantum equivalent of a bit in conventional computing, and the entanglement of qubits is used in performing various sorts of calculation in interesting new ways.
Quantum entanglement is also behind the idea of 'quantum teleportation', by which it is possible to make an exact copy of a particle at a remote location, destroying the state of the original in the process. Over-excited science correspondents have often claimed that quantum teleportation should some day lead to Star Trek-style matter transporters, but there is little truth in this; it is only the doctrine that two indistinguishable quantum entities are really the same that has allowed it to be called teleportation at all, and the technique relies on having a particle very much like the one to be transported already present at the destination. The whole problem of re-assembling at a remote location a pattern of particles matching those found in a person or object remains as intractable as ever; and even if it wasn't, it is in any case not clear that matching the exact quantum states of the original would be a terribly important part of this process.
Significantly, it has also been argued that entanglement is necessary to explain the results of the classic double slit experiment, which have traditionally (but, it would seem, erroneously) been explained in terms of Heisenberg Uncertainty3. In these experiments, a beam of particles is sent through two slits in a barrier, towards a detecting screen. Thanks to diffraction, the result is a pattern of light and dark fringes, showing that the so-called particles interfere with each other much like classical waves on water. The same pattern is built up even if the particles pass through the apparatus one at a time - that is, it's not just that the particles interfere with each other, but that each particle interferes with itself, in a way which seems to force us to the counter-intuitive conclusion that it passes through both holes at once.
However, if we shine a light near the slits, we will see the particle apparently passing through either one slit or the other. Surely, then, it can't be right to say that the particle is really passing through both slits? Or can it? The thing is, if you shine a strong enough light to see which slit a particle goes through, you destroy the interference pattern; either you can see the interference patterns, or you can find out which way the particle goes, but not both; this is often put forward as a classic example of wave/particle duality.
The traditional explanation of this fact, due to Niels Bohr, is given in terms of Heisenberg's uncertainty principle, which says that knowing the position of a particle means making its momentum more uncertain. To tell which slit a particle has gone through, you need to know its position with an uncertainty smaller than the gap between the slits, which we would usually check by bouncing a photon off it. The photon gives the particle a small 'kick', changing the momentum uncontrollably by just enough that the interference pattern is destroyed.
Or so it was thought, until Gerhard Rempe and his colleagues at the University of Konstantz in Germany proved that the pattern is still destroyed even when the particles are tracked using photons with far too little energy to smear out their interference pattern. They did this by using cold rubidium atoms for their particles, pure laser light for their barriers, and low-frequency microwaves emitted by the atoms themselves to detect them. If it was the uncertainty principle which destroyed the patterns, the researchers claim, the experiments wouldn't have worked; the smearing in the momentum is not sufficient to destroy the fringes in this case.
The physicist Yu Shi, based at Cambridge University, has apparently shown that it is just a fortunate numerical coincidence which allowed the uncertainty principle to apparently explain the two-slit experiment; if the full equations of quantum theory are taken into account, the uncertainty principle is seen to be inadequate. Entanglement, however, does the job admirably; according to Shi, the interference pattern disappears as a result of the entanglement between the diffracted particles and their photon partners.
For some of the history of the idea of quantum entanglement, and more discussion of the controversy around it, please see quantum non-locality.
Footnotes
1'Information Flow in Entangled Quantum Systems': http://arxiv.org/abs/quant-ph/9906007
2'Quantum search algorithm implemented using off-the-shelf optics?', http://www.arstechnica.com/wankerdesk/01q2/quantum-1.html
3'An End To Uncertainty,' Mark Buchanan, New Scientist, 6 March 1999.
This piece is largely adapted from a section of my BSc dissertation on Quantum Entanglement and Causality. The full text of the dissertation (partially re-written for a somewhat less technical audience) can be found at http://fergusmurray.members.beeb.net/Causality.html; its original bibliography, annotated, is at http://fergusmurray.members.beeb.net/Bibliography.htm | 
