Meaning of indistinguishability As a mathematical tool, when dealing with something like a many-particle wave function, I understand well what indistinguishability means. It is encoded in the antisymmetry of the wavefunctions and is everywhere in the formalism. In real life, though, what does this mean?  Say, I am writing this question on the computer, and I can think of the electrons on my body and the electrons on the computer screen. It sounds quite crazy to say that they are indistinguishable, and intuitively, couldn't we "tag" them?  The electrons on the screen and those on my body are clearly distinguishable, aren't they?  More generally, if we have two sets of electrons in two different regions of space, and we brought them close, but between them there's a huge (infinite?) potential energy barrier, it is pretty intuitive that we can tag both groups as being on one side or the other of the barrier. What am I missing and how should indistinguishability be understood as a concept?
 A: 
it is pretty intuitive that we can tag both groups as being on one side or the other of the barrier...

You can count the number of electrons on each side of the barrier, but that doesn't mean you've "tagged" them. The wavefunction is still antisymmetric.
Indistinguishability should be understood (and taught) like this: the model doesn't have any observables tied to individual particles. It only has observables tied to regions of space. You can count the number of electrons in a region, but the question "which electrons are in this region?" is meaningless, because the model doesn't have any observables that would make it meaningful.
Asking about real life doesn't avoid the need for a model. In order to answer a question about real life, we either need to (1) refer to experiments that have already been done, or (2) predict the results of new experiments. Doing (2) obviously uses a model, and good models are already thoroughly tested against (1).
When you use intuition, you're using whatever model is in your head, whether or not you call it a model. If the model in your head says you can "tag" individual particles, then you should replace the model in your head with a better model, because the most accurate and comprehensive models available today don't have any observables tied to individual particles. Like Jaynes wrote on page 83 in Probability Theory: The Logic of Science (2003):

From a mere act of the imagination we cannot learn anything about the real world.

A: Lets try to first appreciate the physics behind the mathematical tool, hopefully giving a deeper motivation for likely familiar stuff.
Quantum mechanics says a single particle does not follow a well-defined path. This means if we have two identical particles and we measure them at one instant, and then measure them at a later instant, we simply can't tell which particle went to what place. e.g. if we abstractly call the particles in the first measurement $A$ and $B$, and the particles in the second measurement $C$ and $D$, we can't tell whether $A$ went to $C$ or $D$, and similarly for $B$.
Thus on a mathematical level, the above reasoning implies that the total wave function has to account for all possibilities, and it should not matter which particle we treat as the first or second particle in a given wave function. In other words the total wave functions $\psi(x_1,x_2)$ and $\psi(x_2,x_1)$ should give the same physical results, i.e.
$$|\psi(x_1,x_2)|^2 = |\psi(x_2,x_1)|^2$$
should hold. This means $\psi(x_2,x_1)$ should only differ from $\psi(x_1,x_2)$ by a phase factor $e^{i \alpha}$, i.e.
$$\psi(x_2,x_1) = e^{i \alpha} \psi(x_1,x_2)$$
and
$$\psi(x_2,x_1) = e^{i \beta} \psi(x_1,x_2).$$
Since the phase factor is supposed to be trivial, we can take the convention that $\beta = \alpha$, thus
$$\psi(x_1,x_2) = e^{i \alpha} \psi(x_2,x_1) = e^{2 i \alpha} \psi(x_1,x_2)$$
implies that
$$e^{i \alpha} = \pm 1,$$
allowing for symmetric and anti-symmetric wave functions (and we refer to the identical particles with symmetric wave functions as bosons, and those with anti-symmetric wave functions as fermions). Thus you see why the underlying physics of quantum theory and measurability, when applied systems of identical particles, forces one to consider wave functions with certain symmetry properties under interchange.
If we actually physically stuck a sticker on the particles, obviously the system is really 'particle + sticker' which is not just 'particle', i.e. it's impossible to actually label identical particles and have the system remain as a system of identical particles, and we can't even 'theoretically' label the particles because we can't measure how they move along their paths (this is of course the drastic claim of QM, contrasting with the classical case which says we can).
When you consider the electrons on the screen vs your body you also have to account for all the other particles involved and all the force laws between them and furthermore, for a system with that many particles, you're talking about invoking statistical laws i.e. thermodynamics which gets into why the two systems are more or less independent until you physically interact etc...
If you consider a system of identical particles separated by a potential barrier, it comes down to solving the Schrodinger equation to determine the possible behavior's of the system, and one gets into whether 'quantum tunneling' is possible or whether you just have two independent subsystems of a larger system (remember the total wave function of independent systems just splits into a product).
