As the other answer conveyed (and to which I am just trying to provide some clarification), post-selection is about just looking at a subset of possible measurement outcomes. To my mind, this falls into two different cases, as below. Yes, they are different aspects of the same thing, but they are used very differently by two different communities.
You do some experiments, but you only gather data when certain conditions are fulfilled. Generally it's used to compensate for heralded experimental imperfections (i.e. something is triggered that tells us we've had a undesired result before proceeding with another part of the experiment). For example, you might be using a pair of photons as information or entanglement carriers, but sometimes those photons get lost on the way. If you only do things when both photons are detected, you are post-selecting on their successful arrival.
This is a thought experiment of "how much more powerful could my computer be if I could choose the outcomes of measurements?" and is not a practical proposition.
As a simple example, think about quantum teleportation. In the normal scenario, Alice and Bob share a Bell pair, and Alice has a qubit in an unknown state that she wants to teleport to Bob. She performs a Bell measurement on her two qubits, and sends her measurement outcome to Bob. If Bob is far away from Alice, the information on the measurement result takes a finite time to get there, and it's after that time that he can be thought of as having received the qubit (because he has to compensate for effects of the different results on the qubit he holds).
However, if Alice can post-select on the measurement result as always being one particular result, and Bob knows that she's going to select that one, then Alice doesn't need to send the measurement result to Bob. He can use the qubit he holds immediately. Even stronger, he can use it before Alice has made the measurement, secure in the knowledge that she will! So, not only are you achieving faster-than-light communication, you're actually communicating backwards in time! You can start to imagine how immensely powerful that could be for a computer (compute for an arbitrarily long time and then send the answer back in time to the moment the question was asked).