What causes "spooky force" between entangled particles?
The laws of quantum mechanics - which govern the world in a very small (atomic) scale, are very different than the classical laws of physics that determine the behaviour of macroscopic systems (namely, systems of our scale). Some of these laws can be very confusing and counter- intuitive.
One good example is the concept of entanglement. In classical mechanics, we can treat each object as individual, and know exactly all of its properties. However, in quantum mechanics, there exist systems which are entangled: namely, the properties of one particle depend on the properties of another particle. This can exist, for example, when a sub-atomic particle decays into other two particles. Conservation laws (e.g., of energy, momentum and angular momentum) dictate that the properties of one daughter particle depend on the properties of the other daughter particle. Good example is spin, or intrinsic angular momentum. If the parent particle had spin 0 (e.g., a photon) while the daughter particles have half integer spin (such as electrons), than one of the daughter particles must have spin +1/2, while the other spin -1/2.
Another very important concept of quantum mechanics, that has no classical analogue, is the idea that a particle can only be described in a probabilistic way, until we actually measure its properties. In the above example, we cannot tell which of the daughter particles have a spin 1/2, until we actually measure its spin. Before we do the measurement, the particle behaves as if it has both +1/2 and -1/2 spins combined! This is in strike difference to the classical world, in which one of the particles already have all its properties (spin, in our example) regardless of whether we measure it or not. In quantum mechanics, in strike difference, each of the two particles have both spins - until a measure is actually made. Only when a measure is made, the particle "decides" what path to take; this is known as "collapse of a wave function".
Now, here comes the catch. Suppose that we make such a system in the lab. We do not measure the spin of any particle, but let them propagate away of each other. After they travel some distance, then we measure the spin of one of the particles, and we get a result. Due to the entanglement between the particles, once we know the spin of one of the particles, we immediately know the spin of the other one as well. However, assume that the particles have travelled far enough from each other. According to relativity, no signal can travel faster than the speed of light, and therefore there is no way that the first particle could "tell" the second particle that its spin was measured instantaneously; there is a minimum time it takes the signal to travel between the particles. Thus, we arrived at a paradox, namely an inconsistency between the laws of quantum mechanics (which state that once we measure the properties of one particles we know the properties of the second one immediately), and the laws of relativity, which state that there is a minimum time that needs to be elapsed until the information on our measurement can reach the second particle. This paradox was noted by Einstein (with Boris Podolsky and Nathan Rosen) already in 1935 and is known as the "Einstein-Podolsky-Rosen" (EPR) paradox.
Einstein believed that this paradox implies that there must be something wrong in the laws of quantum mechanics, and that it must be incomplete. He then coined the term "spooky action at a distance" to describe his dissatisfaction from the apparent contradiction between relativity and quantum mechanics that results from the quantum entanglement.
However, numerous experiments since then showed that the laws of quantum mechanics are correct, and therefore the entanglement does occur instantaneously. This seemingly paradox is solved, as it was proven that this effect cannot be used to transmit any information at speeds which are faster than light.