"Quedar" (stay) + "se" (reflective proposition) means with oneselfe or anybody. "Quedarse solo" (being alone) "Quedar con alguien" (meet someone)

"quedarse" has different meanings depend of the context... have a look at this link, for me wordreference is the best online dictionary for resolving my doubts http://www.wordreference.com/es/en/translation.asp?spen=quedarse You can check all differents conjugations form for all spanish verbs.