Quick answer
quedar = to arrange to meet, to be left over, to suit or fit.
quedarse = to stay, to remain in a place or a state.
If you mean "stay", you need the pronoun. Without it you are making a plan.
Side by side
Quedé con Ana a las seis.I arranged to meet Ana at six — no pronoun, it is a plan.
Me quedé con Ana toda la tarde.I stayed with Ana all afternoon — the pronoun turns it into staying.
Solo quedan tres entradas.Only three tickets are left — quedar for what remains.
Memory trick
Pronoun means you stop somewhere. No pronoun means you either made a plan or something is left over.
Typical mistakes
Quedo en el hotel esta noche.
Me quedo en el hotel esta noche.
Staying somewhere needs the reflexive. Without it a Spanish speaker hears that you arranged to meet at the hotel.
Me quedo contigo el sábado a las cinco.
Quedo contigo el sábado a las cinco.
Setting up a meeting is plain quedar. The reflexive version would mean you plan to remain with them.
Check yourself
You want to say you are staying home tonight. Which is right?