It means 'me puedes llamar de tú'. Quiere decir que en vez de usar la forma "usted" para dirigirte a esa persona, uses el 'tú',que en vez de usar la tercera persona, uses la segunda, que es la que utilizarias si hablaras con un amigo o conocido.

It means "me puedes tratar de tú". So you can talk to that person using "tú" instead of "usted".

basicamente es cuando hay mas confianza, es una manera de comunicarte mas informal, por ejemplo: ¡hola! ¿Como estas? (TU) o ¡Hola! ¿Como esta? (USTED)

It means that the other person is giving you a chance to treat him or her with more confidence.

in spanish you can use "tu" o "usted", so "tutear" it's talk with "tu", but when you try to translate that the result will be "you" example: come here! in spanish you can say..... "ven aqui" ( tuteando) or "venga aqui" (usteando) When do you use that? in some places its nomral talk with "tu" (like cartagena, barranquill in colombia) and in other places like bucaramanga, cucuta (colombia) the people usually talk with "usted" in other places like medellin(colombia),quibdo(colombia),argentina you can use "vos" .for example: Do you have hungry? 1.(vos)... vos tenes hambre? 2.(tu)... tu tienes hambre? 3.(usted)... usted tiene hambre? so "me pueden tutear" is the number 2