Stan Thompson is Amelia Richardson's father.
Stan Thompson is first shown paying his daughter, Amelia, a visit. At the time, Amelia has just moved in with Sam Winchester. She warns him in advance that her father can be quite possessive over her and will likely be very demanding and ask many questions. During the whole afternoon, Sam is questioned about why he left Stanford and what he has experienced in his life (partly because he has 'the look' of people that have seen a lot of crap in their lives that they can't forget). He even insults John Winchester by criticizing his post in the marine. Evidently Sam does not like his critical remarks.
Later on, Sam overhears Stan and his daughter talk about her new life. Stan begs Amelia to come home, but she refuses. He says that this life won't last because Sam is a mess, but Amelia contradicts, saying she is a mess too and that she is happy when she is with Sam. Afterwards, he helps Sam by drying the dishes, and they bond over Sam's (former John's) car, The Impala.
At the end of the episode, Stan, Sam and Amelia are talking and laughing about a story Stan's telling, when the phone rings. Amelia walks away to answer it, and Sam tells Stan about losing Dean. He offers his condolences, but then Amelia comes back and tells them that Don, her husband who she thought had died in Afghanistan, is alive.
Stan is not seen again after that.
For the most part, Stan was a seemingly laid-back kind of fellow, but could easily get defensive and protective of his daughter. Like the standard father, he wanted to make sure his daughter was properly taken care of.