Spatial Wavefunction Symmetries and Identical Particles
When we talk about exchanging electron $i$ with electron $j$, we are actually changing the wavefunction according to

$$\Psi(..., x_i, ..., x_j, ...) \to \Psi(..., x_j, ..., x_i, ...).$$

The operation is taken by the parity operator $P$. Applying it twice would
return the wavefunction to its original form. So the following eigenvalue equation is satisfied

$$ P^2 \Psi = 1 \Psi. $$

Then the $\lambda$ eigenvalue of the $P$ operator ($P\Psi = \lambda \Psi$) 
also satisfies an analogous equation 

$$ \lambda^2 = 1. $$

$\lambda$ can only be either $\pm 1$. If it is positive the wavefunction is symmetric under swapping and if the eigenvalue is negative the wavefunction is antisymmetric under swapping. It can also be said that the symmetric (antisymmetric) wavefunction has even (odd) parity but do not confuse it with spatial symmetry.

It is assumed that fermions must be described by antisymmetric wavefunctions
and bosons by symmetric wavefunctions. Many have argumented this fact alluding
to the spin-statistics theorem but others have, as well, stated that such justification is not satisfactory. Besides, it is not completely clear why.