I know this question is going to be borderline philosophy, but please humor me. What is the significance oh identical particles? I can understand that two particles can be identical, but then it actually affects the probabilities which seems to be a fundamentally amazing thing. What are some example where the identical-ness of particles affects probabilities, and given those situations can someone explain why it makes sense? It's just really strange to me that in a situation with two electrons we act as if, and indeed they are, identical. For instance, can someone explain the statistical properties section of the following?: http://en.wikipedia.org/wiki/Identical_particles#Statistical_properties
It is borderline philosophical, but it's also very important for physics and statistics. The classic question is: calculate the probability of the numbers 2 through 12 coming up when you roll two dice: one red and one green. Since clearly Red-3, Green-4 is different from Red-4, Green-3, you calculate the probability distribution. Now, what if both dice are Red, with nothing to tell them apart? Then Red-3,Red-4 is the same as Red-4,Red-3, so the probability distribution should be different! (Calculate as a homework problem.)
There is no way to "prove" that two dice are distinguishable, but every experiment ever done with macroscopic objects such as dice is in agreement with the claim that they are distinguishable.
Moving on to elementary particles, "distinguishable" more or less means "have at least one different quantum number or energy state." It turns out that certain particle interactions lead to scattering distributions consistent with the particles being indistinguishable, while other interactions are consistent with the particles in question being distinguishable. There's lots more interesting stuff on this topic if you read about superconductors or superfluids.
To complement what Carl said, you have the issue of tracking. If you do an experiment with ideal bearing balls, even if you cannot tell them apart, classically you can keep an eye on them, follow their trajectories, and know which one is which. In other words: you can label them.
In QM you cannot continuously observe a system without "destroying" it at every moment. If you throw electron A and electron B at each other, and get two electrons out, you cannot tell which one is which, or if they are actually the same ones: they could have reacted, formed other particles, decayed... When you plug all this into your maths, you get different scattering patterns that can be checked by experiment.