Actually, it means they were born to parents that weren't married.

A bastard is a child who was born out of wedlock, and, in previous eras, legally didn't have a father due to this. This meant that their last name had to be derived from their mother's last name. As a result, bastard children did not have the social power or safety that their wedlock-born peers would have had, and in some places could not be left property or funds by their father. However, a father of a bastard COULD choose to "legitimize" a child, which translates to admitting that the child was his. This would give the child the right to not only use their father's name, but give them the ability to access the social networking and power connected to their father's name, as well as enabling the father to name the bastard child as his heir over his wedlock born children. That said, attempting to request that a child be legitimize could be very dangerous, as the father -especially if they were a noble or otherwise in a precarious but powerful position- could have decided to handle the issue by killing both mother and child, paying the mother to keep her silence, paying the mother to 'dispose' of the child, or (if the child was an infant) kill the mother and pass the child off as being born in wedlock between himself and his wife. Nowadays, it's really more of a petty insult that doesn't have the same social oomph behind it, and the force behind using it as an insult is almost solely dependent on the bastard child's insecurities about their parentage.

As a bastard, I will say it really isn't all that bad nowadays, of course my parents got married shortly after I was born, but people will give you crap for it, unlike what your bully did, it should really just be laughed off, as pointed out prior, it's such a stupid and petty insult, but it gets under people's skin