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Fock space is the Hilbert space of quantum field theory, built as the direct sum of n-particle Hilbert spaces for all n = 0, 1, 2, ... It accommodates states with any number of particles, including the vacuum. Creation operators add particles; annihilation operators remove them; the number operator counts them.
In ordinary quantum mechanics, if you have N identical particles, you work in the N-particle Hilbert space H_N. The wave function has 3N spatial arguments (for three dimensions), and N is fixed throughout the problem. This works well for non-relativistic systems, but it fails for relativistic ones: Einstein's E = mc2 means that collisions with enough energy can create new particles, and particles can annihilate in pairs. You need a framework where the number of particles is a dynamical variable, not a fixed parameter.
Fock space provides this framework. It is defined as the direct sum F = H_0 + H_1 + H_2 + ..., where H_0 = C is the one-dimensional vacuum sector (just the number |0>), H_1 is the one-particle Hilbert space, H_2 is the symmetrized (bosons) or antisymmetrized (fermions) two-particle space, and so on. A general state in Fock space is a vector with components in every sector. The creation operator a_p-dagger maps from H_n to H_{n+1} by adding a particle with momentum p; the annihilation operator a_p maps from H_n to H_{n-1} by removing one. The vacuum is defined by a_p|0> = 0 for all p — there is no particle to remove.
The particle interpretation emerges from the number operator N_p = a_p-dagger a_p, which counts the number of particles with momentum p. The total number operator N = integral N_p d3p / (2pi)3 counts all particles regardless of momentum. For free fields, N commutes with the Hamiltonian, so particle number is conserved — this is why free particles do not spontaneously appear or disappear. Interactions break this: an interaction Hamiltonian like lambda phi4 contains terms that create and destroy particles, and the number operator no longer commutes with H. Particle number is then not conserved, and processes like pair creation and annihilation become possible.
Fock space also makes the statistics of identical particles automatic. For bosons, the commutation relation [a_p, a_q-dagger] = (2pi)3 delta^3(p-q) ensures that multi-particle states are symmetric under particle exchange. For fermions, the anticommutation relation {c_p, c_q-dagger} = (2pi)3 delta^3(p-q) ensures antisymmetry. The Pauli exclusion principle -- no two identical fermions in the same state -- follows from (c_p-dagger)2 = 0. You do not need to impose symmetrization or antisymmetrization by hand; it is built into the algebra of the creation and annihilation operators.