The asteroid belt lies between the orbits of Mars and Jupiter, straddling the inner and outer solar system. Dawn will visit Vesta and Ceres, in the asteroid belt. Dawn's timeline for reaching those targets is shown below.
The inner solar system orbits (in enlargement, at top) are, in order from the Sun: Mercury, Venus, Earth, and Mars.
Jupiter is part of the outer solar system. The outer solar system orbits are, in order: Jupiter, Saturn, Uranus, Neptune, and Pluto. The Kuiper Belt, a realm of icy, rocky objects that often become comets, begins near the orbit of Neptune and continues beyond Pluto.