René Picard was the son of Robert and Marie Picard. He was the nephew of Starfleet Captain Jean-Luc Picard through his father Robert.
Like his famous uncle, René had an interest in space exploration. Jean-Luc and René had a particularly close relationship, as a running joke between the two, Jean-Luc referred to René as "uncle" and René referred to Jean-Luc as "nephew". (TNG: "Family")
René Picard in the Nexus
Tragically, René died in a fire in 2371 at the family home, along with his father. He was survived by his mother. (Star Trek Generations)
Like his father and uncle, René would've suffered from Shalaft's Syndrome. (Star Trek Nemesis)
Apocrypha
In the TNG Relaunch novels, Picard and Doctor Beverly Crusher have a son who they name Rene Jacques Robert Francois Picard, after the cousin and uncle he will never know (Paths of Disharmony).