Permittivity is a measure of how easy or difficult it is to form an electric field inside of a medium. Likewise, permeability is a measure of the ability to form a magnetic field inside of a medium. (Wikipedia: permittivity, permeability)
Permittivity is determined by how much a medium can polarize in response to an electric field.
In cgs units, ε is unitless. In SI units, ε is in units of Farads/meter. This highlights the fundamental difference between the two measures of permittivity in the different unit systems.
In SI units, ε0 is the permittivity of free space and has the value ε0 ≈ 1.85 × 10-12 Farads/meter.
Other materials, specifically dielectrics, have a permittivity that is related to their electric susceptibility, χe, which is a measure of how polarizable a material is in the presence of an electric field. The relationship is given by P = χeE, where P is the electric dipole moment per unit volume of the dielectric. The permittivity is given by:
ε = ε0(1 + χe) SI
ε = (1 + 4πχe) cgs or Gaussian
In SI units, the quantity (1 + χe) is also called the relative permittivity, εr. A value called the dielectric constant is defined as k = εr = ε/ε0. ε in cgs units is the same dimensionless quantity as k in SI units.
Physically, if a dielectric (permittivity doesn't really apply to conductors) with a high value of ε or k is placed in an electric field, the electric field will be greatly reduced inside. This relationship is given by,
D = εE
where D is the displacement vector and it is related to the charge density of the dielectric. Dielectrics are very useful in modern electronics as it is possible to to precisely control their permittivity by doping them with other types of materials. Coaxial cables make use of a dielectric between the two conducting lines.