Capacitors are circuit components that store charge and energy. There are two surfaces, which take the shapes of either two plates, two concentric spheres, or two concentric cylinders. One surface is positive, and the other equally charged negative. An electric field in between the two surfaces provides the mechanism for storing energy.
By definition, capacitance is the ratio of the charge on one of the surfaces, Q, divided by the voltage difference between the surfaces, V. We have C = Q/C. If you want to see how we do this for spheres and cylinders, check out this video.
A dielectric is an insulating material that fills the gap between the charged surfaces, and essentially all capacitors in your electronic gadgets have dielectrics. Dielectrics will partially polarize, produce a small opposing electric field due to the polarization, and weaken the net E-field of the capacitor. This reduces the voltage difference, which finally increases the capacitance. C_new = KC_o. K is a dielectric constant > 1.
Check out the video, which also shows how to calculate the new capacitance when the gaps are partially filled with dielectrics, at least for parallel plate capacitors.