To answer with an example, a lot of these experiments entangle the "spin" property of two particles. In a two-state system such as spin up and spin down, it is tempting to call, say, "up" 1 and "down" 0, but you won't get too much use out of that. What you'll often read is that when you take two entangled particles and take them to opposite ends of the room (or the universe), and you measure your particle as having spin up, you know at once that the other particle must have spin down. That is a function of conservation of angular momentum, and has a very intuitive classical analog.
Before you make a measurement, though, the particle you measured is in a combination of both states. It's not that you don't know which it is; it really is neither (to the best of modern interpretations). If it has equal probability spin up or down, you are equally likely to measure either one. Once you measure it once, though, and get either up or down, if you immediately measure again, you'll get the same result. That's the hastily abused "collapsing the wave function." If you consider the wave function of the particle with regard to its spin states to be representative of a probability distribution of measuring either state, once you measure it, the state you measured has probability one and the rest have probability 0. The new wave function is just a delta, and has been collapsed.
Now to get on with answering your question-- you've not only collapsed this particle's wave function; you've collapsed the other one's too.Instantly. So, you know what the other guy taking measurements on his will measure. What you have no control over is the information content itself. You can't spin your particle so the other one spins in the opposite way. It just doesn't work that way. Information propagation in relativity comes with a caveat-- causal information travels no faster than the speed of light. Non causal information can travel as fast as it likes. It's not spooky action at a distance because it's not really action. Nothing's different in the tangible world as a result of it, which is why you can't use it to send bits. If you're really interested, I recommend Griffiths' Intro to Quantum Mechanics. It was used in all of my related courses, and the author has a way with the inexplicable.