As there is a potential across the plates of a capacitor, there is an associate electric field present. With parallel plates, the electric field lines are generally parallel to each other and at right angles to the plates.
Capacitors require some form of insulator between the two plates, otherwise the charge could not remain on the plates, it would dissipate through the medium between the two plates.
Whilst air is a good insulator, often the capacitor plates need to be kept apart by some form of rigid insulator.
The material between the two plates is called the dielectric. This not only acts as an insulator, but it also determines many of the other properties. A measure known as the dielectric constant affects the level of capacitance achievable for a given capacitor plate size and spacing.
High levels of relative permittivity / dielectric constant can increase the capacitance many times.
The topic of relative permittivity and dielectric constant, etc, is a topic in its own right, and although easy to comprehend, possibly needs to be looked at separately.