Quick answer
volver = to come back, to go back — you move.
devolver = to give something back, to return an object to its owner or shop.
volver a + infinitive is a bonus: it means to do something again.
Side by side
Vuelvo mañana. / Devuelvo el coche mañana.In the first you come back; in the second the car goes back.
Vuelve a llamar.Call again — volver a plus infinitive is the everyday way to say "again".
Te devuelvo la llamada.I'll call you back — devolver, because the call is the thing handed back.
Memory trick
If a thing goes back, use devolver. If a person goes back, use volver. <i>Volver a</i> plus infinitive always means "again".
Typical mistakes
Tengo que volver los libros a la biblioteca.
Tengo que devolver los libros a la biblioteca.
The books are handed over, so it is devolver. Volver never takes an object like this.
Devuelvo a casa a las nueve.
Vuelvo a casa a las nueve.
You are the one moving, so it is volver. Devolver would need something you hand back.
Check yourself
Fill in: "Mañana tengo que ___ estos zapatos a la tienda."