there might be some slight spoilers here but everything i'll say has happened prior to the events of the series and doesn't reveal much
yes, robert is part of a family, the Baratheons. he and Ned were both fosters of Jon Arryn, the lord of a place called The Vale of Arryn (he was also Robert's hand when robert became king). Catelyn Tully, ned's current wife, was slated to marry Ned's older brother, Brandon, and Robert was supposed to marry Lyanna
basically what happened (if memory serves) is that the previous Targaryen king, Aerys II, was pretty much bat-sheet crazy and killed Ned's brother and his father, Rickard, and at some point Aerys' son Rhaegar nabbed Lyanna as well, provoking Ned and Robert into rebellion. as the show has already explained, Lyanna died in some way that is known only to Ned (well and one other guy as well but that's not at all important for now), and Robert killed Rhaegar in a legend-inspiring battle at a place called the Trident
the Lannisters figure in in kind of a complicated way but the important thing to know for now is that Jaime Lannister was a member of the Kingsguard (a group of seven elite bodyguards for the king) and through some chain of events, he killed the Mad King, Aerys II.
there's more to the story as far as what happened with Viserys and Dany, as well as some other members of the Targaryen family (including some people from Dorne, another part of Westeros), but for now all you really need to know is that Vis and Dany were snuck across the narrow sea and live in exile up until the series starts
after re-reading that, it's basically spoiler free, and most of it has been said in the show already. if you have any more questions i'll be glad to answer them