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Particles with integer spin (0, 1, 2, ...) are bosons with symmetric wavefunctions; particles with half-integer spin (1/2, 3/2, ...) are fermions with antisymmetric wavefunctions. This spin-statistics connection is a fundamental theorem of relativistic quantum field theory. Bosons can occupy the same state; fermions cannot. This difference gives rise to dramatically different macroscopic behavior.
From your study of identical particles, you know that quantum mechanics requires a many-particle wavefunction to either stay the same or change sign when two identical particles are exchanged. The crucial result is that this symmetry character is not freely chosen — it is locked to the particle's intrinsic angular momentum, or spin. This is the spin-statistics theorem, one of the deepest results in physics: particles with integer spin are bosons (symmetric wavefunctions) and particles with half-integer spin are fermions (antisymmetric wavefunctions). Electrons, protons, and neutrons all have spin 1/2 and are fermions. Photons have spin 1 and are bosons. Helium-4 nuclei (two protons + two neutrons) have integer total spin and behave as bosons.
The consequences of antisymmetry are profound. For fermions, if you try to put two particles in exactly the same single-particle state, the antisymmetric wavefunction forces the total wavefunction to zero — meaning that configuration simply cannot exist. This is the Pauli exclusion principle, but now you see it is not a separate postulate bolted onto quantum mechanics: it is a direct consequence of antisymmetry. Two electrons cannot share the same set of quantum numbers (n, l, m_l, m_s). This constraint is what forces electrons into successive shells in atoms, gives solids their rigidity, and prevents white dwarf and neutron stars from collapsing under gravity.
Bosons have no such restriction — any number can pile into the same state. This leads to qualitatively different collective behavior. At sufficiently low temperatures, an ideal Bose gas undergoes Bose-Einstein condensation, in which a macroscopic fraction of particles collapses into the single lowest-energy state. Superfluidity in helium-4 and the behavior of laser-cooled atomic gases are direct manifestations of bosonic statistics. Photons, being bosons, populate thermal radiation modes according to the Planck distribution, which you will use when studying blackbody radiation and the photon gas.
The practical dividing line is this: every system made of fermions acquires a kind of rigidity — it resists being compressed into a small number of states — while a system of bosons tends toward coherence and can collectively occupy one state. Chemistry, the periodic table, stellar structure, and the design of lasers all trace back to this single distinction between integer and half-integer spin.