quitar vs quitarse
Quitar is taking something away. Quitarse is taking off your own clothes or stepping aside. The reflexive points back at you.
quitar
To remove, take away or clear something.
Quita eso de ahí.
Take that away from there.
Removing an object from a place.
Me quitó el sitio.
He took my seat.
Taking from someone. The person who loses is the indirect object.
Quita la mesa, por favor.
Clear the table, please.
Quitar la mesa = to clear the table. Poner la mesa = to set it.
quitarse
To take off your own clothes or accessories, or to get out of the way.
Quítate los zapatos.
Take your shoes off.
Clothing: the opposite of ponerse.
Se quitó el abrigo.
He took off his coat.
Body part or clothing comes after the reflexive pronoun.
Quítate de en medio.
Get out of the way.
Idiom, informal. Quitarse de + place.
Tricky bit
'Quitar los zapatos' means remove someone else's shoes. 'Quitarse los zapatos' is the normal way to take your own off. Don't use 'quitar' for yourself.