What is bond polarity and why does it exist?

To understand bond polarity, you first must understand electronegativity. Electronegativity is the ability of the nucleus of any atom to attract electrons from another atom, it is dependant on the nuclear charge on any nucleus and also on the shielding power of surrounding electrons. For example, Fluorine is the most electronegative element in the periodic table because it is late in the second row so has a high nuclear change. Furthermore, because it is in the second row, there are only a few shells surrounding it so the nuclear charge is weakly shielded and will be very strongly attracted to any other electrons.
This then leads to the phenomena known as bond polarity. For example, in water the oxygen is more electronegative than hydrogen due to the reasons discussed above, this will mean that the electrons in the bond are not shared equally between the two nuclei. Instead, the Oxygen pulls the electron closer to its nucleus, this increases the electron density on Oxygen, making the bond polar.