When someone states that they were born "on the cusp" they mean that at the time of their birth, the Sun was very close to making a change in sign -- either slightly before or after the official transition.  Rather curiously, for some unknown reason, it seems to have become quite fashionable to have been born "on the cusp".

Practical interpretations of being "on the cusp" often mean that rather than having a relatively "pure" Sun-sign manifestation exhibited in their personality, the person may show quite demonstrable blend of characteristics of both of the signs on either side of the cusp.

For example, a baby born September 22, 2001 before lunch (more precisely, before 4:05 pm Arizona Time) would have been technically a "Virgo" by Sun sign, while another child born later that very same evening would have a "Libra" Sun-sign.  

The baby born before lunch would be expected to be a fully-developed Virgo, but would almost assuredly also possess some traits more commonly associated with Libra.  Conversely, the baby born that evening would have some residual Virgo tendencies and the Libra energies (although likely slightly prominent) would certainly not be fully developed.

In an extreme example, even identical twins could possess different Sun signs if one twin is born on either side of the exact time of the change of Sign.     

To put it another way, astrology isn't always black and white, but considerable gray areas often blur interpretive connotations.  This is why the positions of all the other planets and points are also so very important.  In the identical twin case, the difference in Sun sign may actually be of relatively small importance when considering the overall Horoscope but would likely identify suggested differences between these two otherwise "identical" twins.