There is no "force" that prevents the electron from falling into the atom's nucleus. Rather, this is a consequence of the modern model of the atom. We are all familiar with the schematic drawing of an atom: electrons orbiting the nucleus much like the planets orbiting the sun. This representation reflects the understanding of physicists as it was until 1911 and is often called "the planetary model of the atom" or the Rutherford Model. This model had to be modified since already in the mid 19th century it was realized that an accelerating charged particle (despite its speed not changing, only its direction) should emit radiation and fall into the nucleus.
The planetary model of the atom
The Bohr Model of 1913 suggested that electrons move in specific paths (i.e. not all radii and not all velocities are allowed) for which the centripetal force is equal exactly to the gravitational pull of the proton. Each path is characterized by its own energy level, and when an electron "jumps" from a high energy level to a lower energy level it loses energy that is equal to the difference between the two levels.
Niels Bohr's model of the atom. n is the energy level and ΔE = hν is the difference in energy between levels, where h is the Planck constant and v is the frequency of the radiation emitted/absorbed when an electron jumps between the levels.
The total energy at a certain radius (kinetic energy resulting from the motion and potential energy resulting from the gravitational pull of the proton) is negative and inversely proportional to the path's radius. Zero energy describes a case in which the electron is ionized (completely separated from the proton, similar to a satellite escaping its orbit around a planet). The first energy level occurs at the smallest radius, known as the Bohr radius (0.0529 nanometers). Smaller radii are simply impossible.
The Bohr model was very attractive in his days and could even explain the band pattern in the emission spectrum of the hydrogen atom. Nevertheless it does have its flaws, the major one being its contradicting the uncertainty principle of 1925. According to Bohr, an electron's velocity and location (radius) can be determined precisely. However, according to quantum theory (which is of course the most acceptable theory since 1927), one can only determine the probability of an electron being located at a certain point in space. These probabilities give rise to atomic orbitals, which are a generalization of Bohr's "paths". Some orbitals do indeed take into account the possibility of an electron being found in the nucleus, but the average radius is that of the lowest energy orbital, and that is the Bohr radius. For higher energy orbitals the average radius is greater.