I'll explain the fundamental physics a bit first. In the last few sentences I explain the difference between the two statistics.
According to quantum field theory, particles are considered to be excited states of fields. The Hamiltonian of a field theory is similar to that of coupled harmonic oscillators. One has an harmonic oscillator for each possible momentum. The creation and annihilation operator which map an energy eigenstate to a higher or lower energy eigenstate can be interpreted as adding or removing a particle with some momentum.
It turns out that for integer spin fields, one has to demand that the creation/annihilation operators commute while for half integer spin the operators have to anti-commute (i.e. satisfy a relation like A B + B A = 0). This rule ensures that local observables will commute when they are space like ...
A detailed explanation is given.