Firstly consider the properties of entangled particles. When one of a pair of entangled particles has its spin measured along any axis, the other particle is found to always has the opposite spin along the same axis. Should the opposite spin axis be determined at the creation of the particle pair (as might be supposed), then for any measurement direction, the required result would be obtained if the spin rotation along the measurement axis was always the same as the nearest spin axis. However, if readings are taken along two different axes for the two entangled particles, the spin would then be expected to be in opposite directions if the two readings are in the same hemisphere as the spin axis, and conversely in the same spin direction for opposite hemispheres. When the fraction of opposite spins is plotted against the angle between the measurement axes for random axes, a straightforward calculation shows that the fraction of readings with opposite spin should reduce almost linearly with the angle between the readings, that is from opposite every time at zero angle when the axes are the same, to never opposite at 180 degrees when the axes are opposite.
It is postulated by quantum theory that the spin direction is not fixed until the spin of one of the particles is measured, and that the spin axis is then the measurement axis. The requirement that the spins are always opposite when measured along the same axis, requires that the spin axis of the second particle must immediately be fixed as opposite to that of the first particle. This is independent of the distance between the particle pair. As in this case the first measurement axis and the spin axis necessarily coincide, Bell pointed out that this case differs from the fixed axis case, where the second measured axis is not same as the first axis. The distribution with the angle between the readings from the spin axis from quantum mechanics has a cosine form. Thus the state of entangled particles prior to measurement can be deduced from the distribution of opposite spin versus the angle between the spin measurements. Extensive experiments gives a cosine form, showing that the axis is not fixed before the first measurement and that the quantum explanation is the correct one. Although this conclusion is contrary to common sense, it is difficult or impossible to come to any other conclusion which fits the facts.
This property of instant communication of the measurement direction to the other particle at some distant place, gives the basis for a means of communication. Assume that many particles are measured in the same direction. Then that direction can be detected by random sampling at the recipient end. If one axis represents yes and another no, then this forms a basis for instant communication.
Consider now a spaceship travelling near the speed of light. Then the spaceship and the astronauts undergo time dilation from relativity theory, an effect is well proven from satellite time clocks and elsewhere. Suppose the spaceship goes on a 10 year round trip near the speed of light to a star such that the onboard time and the age of the astronauts only increases by 5 years. Let the spaceship carries entangled particles with agreement that a message will be sent after a year of spaceship time. When in Earth time is this message received ?
Evidence from particle collision data finds that short lived particles have a longer half life when travelling at speeds near to the speed of light. This suggests that particle time scales are subject to time dilation depending on their speed and acceleration as predicted by Einstein. If this is the case for our entangled particles in the spaceship, then the message will be received at equivalent times, that is after one year Earth time. However if the particle is not subject to time dilation, the Earth based particles will not receive the instantaneous message until 2 years have passed on Earth. Both possibilities cause problems. Firstly, if the message is received after 1 year, when the astronauts return after 5 years when the Earth has aged 10 years, the space travelled particles will be communicating with the Earth bound particles 5 years previous. Thus the particles could transmit events happening in the future. This would require a deterministic universe. Alternatively, if the spaceship particles are not subjected to time dilation, this implies some universal reference time frame. This is not in accord with relativity, and the determination and properties of this time frame would be of fundamental importance.
For more information on entanglement put "entanglement" into Google search
A simple explanation of Bell's experiment is given in
http://www.mathpages.com/home/kmath174.htm
For a more complete analysis of Bell's experiment see
http://www.mathpages.com/home/kmath521/kmath521.htm
Communication difficulties also occur with a finite propagation speed for gravitation. Consider 2 massive bodies in circular orbit about one another. Such a rotating pair send out gravitational waves, and attempts are being made to measure these waves currently. It may be supposed that gravitational wave propagate at close to the speed of light. However, any finite propagation speed for gravity causes difficulty with the conservation of energy and angular momentum. For our 2 bodies in circular orbit, unless the gravitational speed is instantaneous, there will be an apparent lag in the gravitational position of each of the two bodies when viewed from the other body. In which case, when resolving the attractive force, there will be a small component in the direction of motion for each body, increasing the energy and angular momentum of a closed system. That is the conservation of angular momentum and energy is violated.
A further difficulty arises when considering black holes. If the propagation speed of gravity is similar to that of light, then the gravitational attraction from a black hole will be unable to escape and the interior of a black hole will appear to have no mass. This would have a large effect on the nature and properties of black holes. An explanation is that gravity is not the same as electromagnetic radiation in that its effect is to distort space time. However such explanations do not appear to resolve the issues above unless this distortion occurs instantaniously throughout the universe.
For more discussion put "speed of gravity" into Google search.
The above gravitational problems are presented in more detail in
http://www.metaresearch.org/cosmology/speed_of_gravity.asp